Lutwak's Alexandrov–Fenchel conjecture for affine quermassintegrals

For a convex body $K\subset\mathbb{R}^n$, define

$$\mathcal{I}_{k,p}(K):=\frac{\mathcal{Q}_{k,p}(K)}{\mathcal{Q}_{k,p}(B_K)}=\left(\frac{\int_{G_{n,k}}|P_FK|^p\,\sigma(dF)}{\int_{G_{n,k}}|P_FB_K|^p\,\sigma(dF)}\right)^{1/p},$$

where $B_K$ is the Euclidean ball with the same volume as $K$. Lutwak's Alexandrov–Fenchel conjecture. For all $p\in[-n,0]$,

$$\mathcal{I}_{1,p}(K)\geq\mathcal{I}_{2,p}(K)^{1/2}\geq\cdots\geq\mathcal{I}_{k,p}(K)^{1/k}\geq\cdots\geq\mathcal{I}_{n-1,p}(K)^{1/(n-1)}\geq\mathcal{I}_{n,p}(K)^{1/n}=1.$$

The assertion extends the result proved by Lutwak for $p=-1$ and is posed as an open Alexandrov–Fenchel-type inequality for the full interval $[-n,0]$.