Révész–Sarantopoulos conjecture on the Bernstein factor for convex bodies

Let $K$ be a convex body in a normed space, let $x\in\operatorname{int}K$, and let $B_n(K,x)$ denote the Bernstein factor at $x$ for polynomials of degree at most $n$. Write $w(K)$ for the minimal width of $K$, and let $\alpha(K,x)$ be the associated Minkowski asymmetry parameter. Révész–Sarantopoulos conjecture.

$$B_n(K,x)=\frac{2n}{w(K)\sqrt{1-\alpha^2(K,x)}}.$$

The preceding estimate differs from this proposed sharp formula only by a factor arising from replacing $1+\alpha(K,x)$ by its upper bound $2$. The conjecture is stated for arbitrary convex bodies and is open in the supplied text.