The thesis that truth supervenes on being is that any two possible worlds that differ in the truth-value of a proposition also differ in whether some entity exists or not.
Truth supervenes on being if and only if contingent truth supervenes on being. (Necessary truths trivially supervene on everything.)
A theistic argument for contingent truth supervening on being: If p is a contingent truth that does not supervene on being, then no one can have brought it about that p. For how can someone make such a p be true? One’s making something contingently be thus-and-so is a matter of causing and/or refraining from causing (I make an egg be hardboiled by causing it to heat up; I make it be spoiled by leaving it alone). Causation is a relation. Relations hold only between entities. So, any contingent truths that are brought about by someone must supervene on being. The same is true for contingent truths that are brought about by something. Ergo, if there is a contingent truth that does not supervene on being, it is ultimately lacking in explanation (it might have a constitutive explanation, but that has to stop somewhere, and it can’t stop at being/non-being, so the problem remains). And it is contrary to divine sovereignty that there be contingent truths that are not caused to be such, at least indeterministically, by God. (The latter consideration entails that Molinism is false.) So, contingent truth supervenes on being, because God is the first cause (even if indeterministically) of all contingent truth.