Isomorphism of invariant measures vanishing on trivial structures

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 $\operatorname{Str}_{\kappa}[\operatorname{\mathscr{L}}](X)$, each assigning measure $0$ to every trivial structure. Invariant-measure isomorphism conjecture. There is a measurable isomorphism

$$f:\operatorname{Str}_{\kappa}[\operatorname{\mathscr{L}}](X)\to\operatorname{Str}_{\kappa}[\operatorname{\mathscr{L}}](X)$$

such that $\mu_0$ is the pushforward of $\mu_1$ along $f$:

$$\mu_0= f_*\mu_1.$$

This is presented as a stronger form of the conjectured equivalence between the corresponding random forcing notions.