Betke–Weil reverse Alexandrov–Fenchel conjecture for mixed volumes

Let $K_1,\ldots,K_m$, with $m\leq n$, be compact convex sets in $\mathbb{R}^n$, and let $\alpha_1,\ldots,\alpha_m\in\mathbb{N}$ satisfy $\alpha_1+\cdots+\alpha_m=n$. Write $V(K_1[\alpha_1],\ldots,K_m[\alpha_m])$ for the mixed volume with multiplicities $\alpha_i$, $V_{\alpha_i}(K_i)$ for the intrinsic volume of order $\alpha_i$, and $\binom{n}{\alpha_1,\ldots,\alpha_m}$ for the multinomial coefficient.

Betke–Weil conjecture.

$$\binom{n}{\alpha_1,\ldots,\alpha_m}V(K_1[\alpha_1],\ldots,K_m[\alpha_m])\leq V_{\alpha_1}(K_1)\cdots V_{\alpha_m}(K_m).$$

If $\dim(K_i)\geq\alpha_i$ for $i=1,\ldots,m$, equality holds if and only if $\dim(K_i)=\alpha_i$ for $i=1,\ldots,m$ and the affine hulls of $K_1,\ldots,K_m$ are pairwise orthogonal.

This is a reverse counterpart to the Alexandrov–Fenchel inequality and gives the sharp upper bound suggested by Ulrich Betke and Wolfgang Weil. The supplied text does not provide evidence that the conjecture has been resolved.