Madiman–Wang conjecture on Rényi entropy of sums

For $\beta>0$, let the generalized Gaussian have density

$$G_{\beta,p}(x)=\alpha(1-\beta|x|^2)_+^{1/(p-1)},$$

where $\beta$ and $\alpha$ are chosen so that it integrates to one. Let $X_j$, $j=1,\dots,n$, be independent random variables with densities $f_j$, and let $Z_j$ be independent random variables distributed according to $G_{\beta_j,p}$, where $\beta_j$ is chosen so that $h_p(X_j)=h_p(Z_j)$. Madiman–Wang conjecture. One has

$$h_p(X_1+\dots+X_n)\geq h_p(Z_1+\dots+Z_n).$$

This conjecture asks whether generalized Gaussians minimize the Rényi entropy of a sum among independent random variables with prescribed individual Rényi entropies. It is presented as an open question in the source and generalizes the corresponding Gaussian minimization statement for Shannon entropy.