Kolesnikov–Werner's many-body functional Blaschke–Santaló conjecture

Let $n\ge1$ and $k\ge2$. Let $\rho:\mathbb R\to\mathbb R_+$ be non-increasing, and let $f_1,\ldots,f_k:\mathbb R^n\to\mathbb R_+$ be even integrable functions. For

$$\mathcal{S}_2(x_1,\ldots,x_k)=\sum_{1\le i<j\le k}\langle x_i,x_j\rangle,$$

assume

$$\prod_{i=1}^k f_i(x_i)\le \rho\bigl(\mathcal{S}_2(x_1,\ldots,x_k)\bigr)$$

for all $x_1,\ldots,x_k\in\mathbb R^n$. Kolesnikov–Werner's conjecture. There is a constant $C_k$ such that

$$\prod_{i=1}^k\int_{\mathbb R^n}f_i(x)\,\mathrm d x\le\left(\int_{\mathbb R^n}\rho\bigl(C_k|u|^2\bigr)^{1/k}\,\mathrm d u\right)^k.$$

This extends the two-function functional Blaschke–Santaló inequality to many-body pairwise interactions. It is known for $k=2$, for unconditional functions, and in the Gaussian case (even without symmetry assumptions in the later entropic result), while the general case remains open.