Révész–Sarantopoulos conjecture on the real linear polarization constants

Let $c_n(X)$ denote the $n$th linear polarization constant of a normed space $X$, and let ${\bf R}^n$ be the real Euclidean space. Révész–Sarantopoulos conjecture.

$$c_n({\bf R}^n)=n^{\frac{n}{2}}.$$

This conjecture asks for the exact real analogue of Arias-de-Reyna's complex result $c_n({\bf C}^n)=n^{n/2}$. The paper states that it had appeared previously in work of Ball and Arias-de-Reyna and had also been formulated by Révész and Sarantopoulos; the preceding theorem gives only the bounds $n^{n/2}\leq c_n({\bf R}^n)\leq 2^{n/2-1}n^{n/2}$, so the conjecture remains open here.