Makeev's concurrent-planes conjecture for regular simplices

Let $F:S^{n-1}\to\mathbb{R}$ be an odd function, and let $\Sigma_n\subset\mathbb{R}^n$ be a regular simplex of edge-length $1$, with vertices $v_1,\ldots,v_{n+1}$. For $A\in SO(n)$ and each $1\leq i<j\leq n+1$, consider the $(n-1)$-plane

$$\{x\in\mathbb{R}^n\mid \langle x,A(v_j-v_i)\rangle=F(A(v_j-v_i))\}.$$

Makeev's conjecture. There exists an $A\in SO(n)$ such that all $n(n+1)/2$ of these $(n-1)$-planes are concurrent. The conjecture is a reformulation of Makeev's universal-cover conjecture in terms of continuous functions and supporting planes; the source gives no resolution.