Eskenazis–Ivanisvili conjecture on Gaussian Markov–Bernstein inequalities

Let $\gamma$ denote the standard Gaussian measure on $\mathbb R$. For $n,d\in\mathbb N$, let $\mathcal P_d(\mathbb R^n)$ be the space of polynomials on $\mathbb R^n$ of degree at most $d$, and let $\nabla f$ be the full gradient of $f$. Eskenazis–Ivanisvili conjecture. For every $p\in[1,\infty)$, there exists a constant $C(p)>0$ such that, for every $n,d\in\mathbb N$, one has

$$\\|\nabla f\\|_{L^p(\gamma^n)} \le C(p)\sqrt d\\,\\|f\\|_{L^p(\gamma^n)} \quad \forall f\in \mathcal P_d(\mathbb R^n).$$

This conjecture asks for a dimension-free Gaussian integral Markov–Bernstein inequality with the sharp square-root dependence on the polynomial degree. The paper states that its conjecture is confirmed for even integer exponents and that improved estimates are obtained for all $p\ge4$; the general assertion as stated is therefore not established here.