Change-of-variable conjecture for the entropic measure under diffeomorphisms

Let $M$ be the underlying multidimensional space, let $\mathcal{P}$ denote the space of probability measures on $M$, and let $\mathbb{P}^\beta$ be the entropic measure with parameter $\beta$. For a $\varphi^2$-diffeomorphism $h:M\rightarrow M$, write $h_*\mu$ for the pushforward of $\mu$ and let $U:\mathcal{P}\rightarrow\mathbb{R}$ be a bounded Borel function. Change-of-variable conjecture. For every $\varphi^2$-diffeomorphism $h:M\rightarrow M$, there exists a function $Y_h^\beta:\mathcal{P}\rightarrow\mathbb{R}$ such that

$$\int U(h_*\mu)\,\mathbb{P}^\beta(d\mu)=\int U(\mu)Y_h^\beta(\mu)\,\mathbb{P}^\beta(d\mu).$$

It suffices to consider functions of the form $U(\mu)=u(\mu(M_1)),\ldots,\mu(M_N))$ for measurable partitions $M=\bigcup M_i$ and bounded measurable $u:\mathbb{R}^N\rightarrow\mathbb{R}$. Moreover, the density should have the form

$$Y_h^\beta(\mu)=\exp\left(\beta\int_M\log\det Dh(x)\,\mu(dx)\right)\cdot Y_h^0(\mu),$$

where $Y_h^0(\mu)$ is independent of $\beta$. The conjecture seeks a change-of-variable formula for the entropic measure on the space of probability measures; the existence and stated factorization of the density are not established in the supplied text.