The multilinear good-fibration Brascamp–Lieb inequality

Let $V$ be a manifold and let $\Gamma_1,\dots,\Gamma_n$ be a transverse family of good fibrations of $V$, with corresponding maps $\pi_j:V\to U_j$ and forms $\omega_j$. For each $x\in V$, let $\Phi_x$ be the map associated with the corresponding flows, and assume that every $\Phi_x$ is injective. Then, for all nonnegative $f_j\in L^1(U_j)$, the multilinear good-fibration inequality.

$$\int_V \prod_{j=1}^n f_j(\pi_j x)^{1/(n-1)}\,\omega_1(x)\wedge\dots\wedge\omega_n(x)^{1/(n-1)}\;{\rm d}x\leq\prod_{j=1}^n\left(\int_{U_j}f_j\right)^{1/(n-1)}.$$

This is a Brascamp–Lieb-type estimate for transverse families of good fibrations under the global injectivity hypothesis on the maps $\Phi_x$. The supplied text does not state whether the assertion is proved or remains open.