The FBI or Interpol should examine the documentation of all their trips to see if any of them correspond to any art thefts in the respective countries they visited.
They obviously know art since they kept the painting rather than put it up for sale......
There has to be a black market for stolen artifacts but I'll be dared if I know how to go about entering it to sell something.
The FBI is busy with the coup. They don’t have time for regular crime.