Extreme-point conjecture for Nikol'skii constants

Let $D\subset\mathbb{R}^d$ be a convex compact set, and let $C(\mathcal{P}_N,D,\boldsymbol{x})$ denote the Nikol'skii constant at $\boldsymbol{x}\in D$ for the polynomial space $\mathcal{P}_N$. An extreme point of $D$ is a point of $D$ that does not lie in any open line segment joining two points of $D$. Extreme-point conjecture. The maximum

$$\max_{\boldsymbol{x}\in D} C(\mathcal{P}_N,D,\boldsymbol{x})$$

is achieved at an extreme point of $D$. The conjecture would sharpen the understanding of where Nikol'skii constants attain their maxima; the surrounding discussion notes that boundary maxima are known for several domains, including balls, ellipsoids, cubes, and affine images of cubes, but does not establish the claim for every convex compact domain.