The generalized Busemann intersection inequality

Let $L$ be a measurable star set in $\mathbb R^n$, let $1\leq j<k<n$, and let $G_{n,r}$ denote the Grassmannian of $r$-dimensional linear subspaces with invariant probability measure $d_*\tau_0$ or $d_*\zeta_0$. Write $V_k$ for $k$-dimensional volume and $\widetilde V_m$ for the corresponding generalized section functional. Then

$$\int_{G_{n,k}}[V_k(L\cap\tau_0)]^{n/j}\,d_*\tau_0 \leq \left(\frac{b_k}{b_j}\right)^{n/j}\left(\int_{G_{n,j}}[\widetilde V_k(L\cap\zeta_0)]^{n/k}\,d_*\zeta_0\right)^{k/j}.$$

Generalized Busemann intersection inequality. More generally, for an arbitrary real number $m$ such that the integral on the right-hand side exists in the Lebesgue sense,

$$\int_{G_{n,k}}[\widetilde V_m(L\cap\tau_0)]^{n/j}\,d_*\tau_0 \leq \left(\frac{b_k}{b_j}\right)^{n/j}\left(\int_{G_{n,j}}[\widetilde V_m(L\cap\zeta_0)]^{n/k}\,d_*\zeta_0\right)^{k/j}.$$

This would generalize the Busemann intersection inequality and the known $k$-plane-transform estimate from convex bodies to arbitrary measurable star sets and from the $k$-section functional to arbitrary admissible real powers $m$. The source presents it as a conjecture and gives no resolution.