Universality of random generic extensions for invariant measures

Let $X$ be a set, let $\operatorname{\mathscr{L}}$ be a language, and let $\kappa$ be the relevant cardinal. Let $\mu_0$ and $\mu_1$ be invariant measures on $\operatorname{Str}_{\kappa}[\operatorname{\mathscr{L}}](X)$, and let $\Generic_0$ be a $\mu_0$-random generic filter over $V$. Suppose that both measures assign measure $0$ to every trivial structure. Random-generic universality conjecture. In $V[\Generic_0]$ there is a $\mu_1$-random generic filter over $V$. This proposes that, under the stated null-trivial-structure condition, one random generic extension contains random generics for every other invariant measure.