Bilinear standard simplex conjecture for positive correlation

Let $m\geq3$, let $0<\rho<1$, and let $(\Omega_i)_{i=1}^m$ and $(\Omega_i')_{i=1}^m$ minimize the bilinear Gaussian noise-stability problem with equal prescribed measures, under the condition $m-1\leq n+1$. Let $z_1,\ldots,z_m$ be the vertices of a regular simplex in $\mathbb{R}^{n+1}$ centered at the origin. Bilinear standard simplex conjecture. There exists $w\in\mathbb{R}^{n+1}$ such that, for every $i$,

$$\Omega_i=-\Omega_i'=w+\{x\in\mathbb{R}^{n+1}:\langle x,z_i\rangle=\max_{1\leq j\leq m}\langle x,z_j\rangle\}.$$

This bilinear positive-correlation formulation is introduced to study the negative-correlation case, where plurality is expected to minimize noise stability. The supplied context does not state a resolution.