The word "liar" is imprecise. We can describe as a liar someone who speaks falsely on occasion, not necessarily in every statement. Probably you meant the indirect question to be "do you always speak falsehood?".

Putting aside what the indirect question is for the moment, if one of B and C was the random one, then A couldn't know whether B and C would give the same answer to a given question on any given occasion. I suspect that asking a question to which the answer could be undeterminable is bending the rules of the challenge, as you are only allowed to ask yes/no questions.

But also you seem to be asking a paradoxical question ("do you always speak falsehood?")which someone who always speaks falsehood would find impossible to answer without ruining their reputation. I think that asking such a question, even in an indirect way, must also be against the rules.

The challenge as worded does not tell us whether the gods are aware of the propensities of the others. However I see that the proposed answer to the challenge is based on the assumption that they do know each other's propensities.