Mahler's conjecture for the restricted class with s-concave power in \mathcal{F}

Let $s\in(-\frac{1}{n},0)$ and let $g:\mathbb{R}^{n}\to\mathbb{R}_{+}$ be even with $g^{s}\in\mathcal{F}$, where $\mathcal{F}$ is the set of convex lower semicontinuous functions $f:\mathbb{R}^{n}\to(0,+\infty)$ such that, for every $x\ne0$, $\lim_{t\to+\infty}f(tx)=+\infty$ and $t\mapsto f(tx)/t$ is non-increasing on $(0,+\infty)$. Restricted Mahler conjecture.

$$\int_{\mathbb{R}^{n}}g(x)\,dx\int_{\mathbb{R}^{n}}\mathcal{L}_{s}g(y)\,dy\geq\frac{1}{1+ns}\,\frac{4^{n}}{(1+s)\cdots(1+ns)},$$

with equality for $g(x)=(\max(1,\|x\|_{\infty}))^{1/s}$. This conjecture is known for unconditional functions, in particular in dimension one, and the paper proves it in dimension two when $-1/s\in\mathbb{N}$; the general statement remains open.