Benítez–Sarantopoulos–Tonge conjecture on the linear polarization constant of Euclidean space

Let $n$ unit vectors $v_1,\ldots,v_n$ belong to $\mathbb R^n$. Define the maximum of the absolute product of their inner products with a unit vector by

$$\sup_{\lVert x\rVert_{\mathbb R^n}=1}\left\lvert\langle x,v_1\rangle\cdots\langle x,v_n\rangle\right\rvert.$$

Benítez–Sarantopoulos–Tonge conjecture. One has

$$\sup_{\lVert x\rVert_{\mathbb R^n}=1}\left\lvert\langle x,v_1\rangle\cdots\langle x,v_n\rangle\right\rvert\ge n^{-n/2},$$

and equality holds if and only if $\{v_i\}_{i=1}^n$ is an orthonormal system.

This conjecture asks for the exact value of the $n$-th linear polarization constant of $\mathbb R^n$, which is known to be at least $n^{n/2}$. The corresponding inequality and equality characterization were posed by Benítez, Sarantopoulos, and Tonge; the parser supplies no evidence of a resolution, so the conjecture remains open.