Rényi entropy power inequality for generalized Gaussians

Let $X_1,\ldots,X_k$ be independent random vectors taking values in $\mathbb{R}^n$, and let $p>\frac{n}{n+2}$. Let $Z_i$ be independent random vectors, each a scaled version of the generalized Gaussian $Z^{(p)}$, such that $h_p(X_i)=h_p(Z_i)$.

Rényi entropy power conjecture.

$$N_p(X_1+\cdots+X_k)\geq N_p(Z_1+\cdots+Z_k).$$

This conjecture refines the entropy-power inequality and would identify scaled generalized Gaussians as the minimizers of the Rényi entropy power of a sum under fixed individual Rényi entropies. It is known for $p=1$, for $p=\infty$ with $k=2$, and for $p=\infty$ in dimension $n=1$ for arbitrary $k$; the general case remains open.