Propeller conjecture

Let $m>3$, let $a_1,\dots,a_m>0$ satisfy $\sum_{i=1}^m a_i=1$, and let $w\in\mathbb{R}^{n+1}$ be the vector defined in the paper. Set $\overline{w}^{(i)}=w/a_i$. Consider measurable sets $\Omega_1,\dots,\Omega_m\subseteq\mathbb{R}^{n+1}$ covering $\mathbb{R}^{n+1}$, with $\gamma_{n+1}(\Omega_i)=a_i$, that maximize

$$\sqrt{\frac{\pi}{2}}\sum_{i=1}^m\left\|\int_{\Omega_i}(x-\overline{w}^{(i)})\gamma_{n+1}(x)\,dx\right\|^2.$$

Assume $m-1\le n+1$. Propeller Conjecture. The maximizing sets $\Omega_1,\dots,\Omega_m$ are simplicial cones over a regular simplex. This conjecture is a variant of problems motivated by kernel clustering and generalized Grothendieck inequalities. The source relates it to the small-correlation limit of the standard simplex conjecture, but does not establish the stated general case.