Maximal restricted sections of the cross-polytope

Let $B_1^n=\operatorname{conv}\{\pm e_1,\ldots,\pm e_n\}$ be the cross-polytope, and let

$$\mathcal{M}=\left\{a\in\mathbb{R}^n:\ \sum_{r=1}^n a_r^2=1,\ \sum_{r=1}^n a_r=0\right\}.$$

For $a\in\mathcal{M}$, consider the central hyperplane section $B_1^n\cap a^\perp$. Maximal-section conjecture. The function

$$a\in\mathcal{M}\longmapsto \operatorname{vol}_{n-1}(B_1^n\cap a^\perp)$$

attains its maximum for $n<6$ at

$$\left(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0,\ldots,0\right),$$

whereas for $n\geq 6$ it is maximized at

$$\left(\underbrace{\frac{1}{\sqrt{n(n-1)}},\ldots,\frac{1}{\sqrt{n(n-1)}}}_{n-1\text{ times}},-\sqrt{\frac{n-1}{n}}\right).$$

The claim concerns restricted sections whose hyperplanes pass through the barycenter of a facet. The supplied text does not state whether this assertion is known or open, so its database status remains open.