Random randomly silent

In a new paper by Gabriel Uzquiano called ``How to solve the hardest logic puzzle ever in two questions" he shows how to solve the puzzle in two questions -- one way uses the ignorance of the gods with respect to Random's future answers (if such there be) and the other way uses potentially unanswerable self-referential questions. In either case, the trick is to get information from a god's inability to answer certain questions. Uzquiano suggests a further amendment to the puzzle to avoid such two question solutions: have Random randomly say 'ja', 'da' or remain silent (instead of silence one could have the god randomly suffer a head explosion -- doesn't matter as the logic is the same).

Puzzle. Three gods A, B and C are called, in some order, True, False and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely or whether Random speaks at all is a completely random matter. Your task is to determine the identities of A, B and C by asking yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for ‘yes’ and ‘no’ are ‘da’ and ‘ja’, in some order. You don't know which word means which.

With the further condition:

* Whether Random answers ‘da’ or ‘ja’ or whether Random answers at all should be thought of as depending on the toss of a fair three-sided dice hidden in his brain: if the dice comes down 1, he doesn't answer at all; if the dice comes down 2, he answers ‘da’; if 3, ‘ja’.

It is left as an open question, whether this puzzle can be solved in two questions. Landon proves that it cannot be solved in two questions here.