Sharp two-summand Rényi entropy power inequality constant

Let $X_1$ and $X_2$ be independent random vectors taking values in $\mathbb{R}^n$, and let $p>\frac{n}{n+2}$. Let $Z^{(p)}_1$ and $Z^{(p)}_2$ be independent random vectors drawn from $g_{\beta_p}$.

Sharp two-summand Rényi entropy power conjecture.

$$N_p(X_1+X_2)\geq C_{p,n}\bigl[N_p(X_1)+N_p(X_2)\bigr],$$

where

$$C_{p,n}=\frac{1}{2}\frac{N_p(Z^{(p)}_1+Z^{(p)}_2)}{N_p(Z^{(p)})}.$$

This proposes the optimal constant in the two-summand Rényi entropy power inequality, with the constant determined by the sum of two independent generalized Gaussians. The source presents this as an aggressive conjecture and does not establish it in general.