Regular simplex conjecture for the spectral gap

Let $\mathfrak{M}_n$ be the moduli space of all $n$-simplices with unit diameter. For $n\geq 2$, let $p_0,p_1,\ldots,p_n\in\mathbb{R}^n$ satisfy

$$|p_i-p_j|=1\qquad\text{for }0\leq i\neq j\leq n,$$

so that these points define a regular simplex. Regular simplex conjecture. The regular simplex uniquely minimizes the gap function on $\mathfrak{M}_n$. The conjecture concerns the unresolved existence and identity of a gap-minimizing simplex; difficulties include the behavior of gaps for families of collapsing simplices.