No…but there may have been attempts to resolve this
Lao Che makes remarkably few appearances in Indiana Jones titles, with none of them taking place after Temple of Doom. Wikipedia does have an unsourced claim that he was considered to reappear in what became Indiana Jones and the Kingdom of the Crystal Skull before the actor's death.
Before the death of Roy Chiao, Steven Spielberg was planning to return Che in Indiana Jones and the Kingdom of the Crystal Skull, but he declined the idea after Chiao's death in 1999. Also, Ric Young, the actor that portrayed Kao Kan, expressed interest in reprise his role of Kan in the movie, but he declined after Roy's death.
(I've asked "Did Spielberg plan on having Lao Che return in Indiana Jones and the Kingdom of the Crystal Skull?" over on Movies & TV Stack Exchange, to try to find information that might corroborate this).
The promotional trailer for Indiana Jones and the Staff of Kings shows Indy drinking at "Lao Che Cocktails" before two thugs approach saying "I heard you were looking for the boss", suggesting that he was trying to meet Lao Che.
In the final game, the Lao Che Lounge is a location Indy visits, but the man himself does not appear.