Wang–Madiman generalized Gaussian maximizer conjecture for Rényi entropy power

For $p>\frac{d}{d+2}$, define $\beta_p$ by

$$\frac{1}{\beta_p}=\frac{1}{p-1}+\frac{d+2}{2}.$$

Let $Z^{(p)}$ be a random vector drawn from $g_{\beta_p}$. Let $X_1,\ldots,X_n$ be independent random vectors in $\mathbb{R}^d$, and let $Z_i$ be independent random vectors, each a scaled version of $Z^{(p)}$, satisfying $h_p(X_i)=h_p(Z_i)$. Wang–Madiman's conjecture.

$$N_p(X_1+\cdots+X_n)\geq N_p(Z_1+\cdots+Z_n).$$

The generalized Gaussians arise as maximizers of Rényi entropy power under a variance constraint. The conjecture asserts that they also minimize the Rényi entropy power of sums among independent summands with matching Rényi entropies.