A faint voice comes through the crackled phone line. On the other end, Tomeu Vadell, speaking from a military counterintelligence prison in Venezuela's capital, asks his daughters in Louisiana whether they've gone to church and says he plans to spend his Sunday doing pushups to keep his body and spirit intact.The call ends abruptly after two minutes, leaving Cristina and Veronica Vadell wondering when they'll next hear from their dad, who along with five other executives from Houston-based Citgo has spent 15 months jailed in Venezuela on what their families say are trumped-up corruption charges. "He always tells us they can take away his freedom but never his dignity," said 27-year-old Cristina, who has followed in her father's footsteps and is an oil engineer in Lake Charles, Louisiana, where she has lived most of her life. As the Trump administration plunges ahead in its effort to unseat Venezuelan President Nicolas Maduro, the fate of the so-called Citgo Six — five of them, like Vadell, American citizens with deep roots in Louisiana and Texas — lies in the balance. As does that of the American company they worked for, which is a major prize in the power struggle between Maduro and a rival the U.S. recognizes as Venezuela's rightful leader: Juan Guaido. Their families complain the men are being held in inhumane conditions, sharing overcrowded basement cells built for 22 people with nearly four times that number of inmates. They say the crowded conditions require the men to sometimes sleep on the floor and go without access to fresh air or sunlight for weeks. Vadell's family says he has lost more than 60 pounds due to malnutrition. In a photo snapped clandestinely with a cellphone last month and provided to The Associated Press, he looks like a prisoner of war with sunken eyes and cheeks, a green army jumpsuit hanging from his gaunt frame. Their case shows no sign of advancing. A preliminary hearing has been postponed 12 times for little apparent reason, leaving the families to question whether their loved ones are being held as pawns in a high-stakes political negotiation. The next hearing date is Wednesday. "The situation, as volatile as it is now, brings more uncertainty," said Cristina Vadell. "We can't predict the future. We don't know what's going to happen. But I do know my father is staying strong for us and we aren't going to give up until we bring him home." The families' saga began the weekend before Thanksgiving in 2017, when Vadell and the other executives got a call from Nelson Martinez, then head of Citgo's parent, Venezuela's state oil giant PDVSA, asking that they travel to Caracas for a last-minute budget meeting. The group flew out on a corporate jet. They included Vadell, vice president of refining; Gustavo Cardenas, head of strategic shareholder relations as well as government and public affairs; Jorge Toledo, vice president of supply and marketing; Alirio Zambrano, vice president and general manager of Citgo's Corpus Christi refinery; Jose Luis Zambrano, vice president of shared services; and Jose Angel Pereira, the president of Citgo.What happened next upended the families' serene lives. A group of armed and masked security agents rushed into a PDVSA conference room and arrested all six executives.Hours later, Maduro's attorney general appeared on state TV announcing charges of embezzlement stemming from a proposal to refinance some $4 billion in Citgo bonds by offering up a 50 percent stake in the company as collateral. "On Monday he left and he was supposed to come back Tuesday," said Vadell's wife, Dennysse. "He went into a meeting and never came back." Then Maduro himself accused them of "treason," though they have not been charged with that crime. The arrests kicked off a purge inside Venezuela's oil industry that a few days later saw Martinez, the PDVSA head, and a former oil minister among dozens others jailed. In Martinez's place, Asdrubal Chavez, a cousin of the late President Hugo Chavez and close ally of Maduro, was named Citgo president. In December, Martinez died in state custody, further alarming the families of the Citgo employees.Citgo, which controls about 4 percent of U.S. refining capacity, has provided almost no support to the jailed executives despite an indemnity agreement that obligates it to act on the men's behalf, current Citgo employee said, speaking on condition of anonymity for fear of being punished by the company. In the ensuing months, Citgo also terminated their pay, the employee said. "The only communication I had with Citgo when this happened was they called to tell me not to go to the media and that they were going to every house to pick up the company cars," said Maria Elena Cardenas, whose husband is among those jailed.A battle is now raging at Citgo headquarters, the employee said, as a result of U.S. sanctions on PDVSA last month that effectively block American companies from buying Venezuelan oil, diverting any payments into an escrow account controlled by Guaido, who the U.S. and dozens of other countries recognize as Venezuela's interim president. Most of the employees loyal to Maduro have left, while any reference to PDVSA has been scrubbed from the company's facilities, along with portraits of South American independence hero Simon Bolivar. Maduro has vowed to defend Citgo from seizure, saying it belongs to the Venezuelan people. His attorney general last week filed criminal charges against the new PDVSA and Citgo boards appointed by Guaido. The U.S. is keeping a tight lid on whatever efforts it has undertaken to help the men. American consular officials have been repeatedly denied access to them in jail because the Vienna Convention doesn't obligate Venezuela to recognize their dual American nationality. U.S. officials have raised concerns in diplomatic notes and meetings with the foreign ministry, said a U.S. official who spoke on condition of anonymity because he wasn't authorized to discuss the case. Secretary of State Mike Pompeo, when asked about the case in a Feb. 6 interview with Fox Business Network, said only that "anywhere there is an American who is wrongfully detained, in this case by the thug Maduro, the United States government is incredibly focused on obtaining their release." Some family members wonder if the U.S. could be doing more.Venezuelan human rights groups don't include the men among the almost 1,000 people classified as political prisoners, and the hushed approach contrasts with the very public diplomatic push that secured the release last year of Joshua Holt, a Utah man who was held for more than two years in a Caracas jail on weapons charges that were also seen as bogus. A Venezuelan official who regularly fielded U.S. complaints in the Holt case said Americans have been largely silent with regard to the Citgo employees. He spoke on the condition of anonymity because he isn't authorized to speak to the press. "We are grateful for the people who continue to help us, but we're not convinced the U.S. government is taking these Americans into account when making policy toward Venezuela," said Veronica Vadell. For Maria Elena Cardenas time is running out. Her 18-year-old son, Sergio, suffers from a rare metabolic disease that has stunted his physical growth. Since his father's arrest, he has been having panic attacks and screaming at night. The two traveled to Caracas recently at great risk to Sergio's health for a two-hour jailhouse visit to calm the teen's nerves."He shouldn't be in jail. He should be home with his us, his family," the younger Cardenas said, his voice quivering with emotion. "He's the bravest person I've ever known. He's the greatest father in the world."