Kalantzopoulos–Saroglou's geometric many-body Blaschke–Santaló conjecture

Let $n\ge1$ and $k\ge2$, and let $K_1,\ldots,K_k\subset\mathbb R^n$ be origin-symmetric convex bodies. Kalantzopoulos–Saroglou's conjecture. If

$$\mathcal{S}_2(x_1,\ldots,x_k)\le C_k$$

for all $x_i\in K_i$, $i=1,\ldots,k$, then

$$\prod_{i=1}^k|K_i|\le|B^n|^k.$$

This is the geometric polarity counterpart of the many-body functional inequality. The source gives related partial results under unconditionality and establishes the conjecture in certain special settings, but the full statement remains open.