Ball–Nayar–Tkocz–Madiman–Melbourne–Xu norm conjecture for Rényi entropy

Recall that a probability measure on $\mathbb{R}^n$ is $\kappa$-concave if it satisfies the Brunn–Minkowski-type inequality associated with $\kappa$; a random vector is $\kappa$-concave when its law is such a measure. For $p\in[0,\infty]$, let $N_p$ denote the $p$-Rényi entropy power. Let $X$ be a $\kappa$-concave random vector in $\mathbb{R}^n$, with $\kappa\in[-\infty,1/n]$. Ball–Nayar–Tkocz–Madiman–Melbourne–Xu conjecture. The function

$$v\mapsto \begin{cases} N_p^{1/2}(v\cdot X),&v\neq 0,\\0,&v=0\end{cases}$$

defines a norm on $\mathbb{R}^n$. This is a proposed reverse Rényi entropy power inequality; homogeneity is immediate, while the conjecture reduces to the triangle inequality and is known in some special cases, including the Shannon case for log-concave vectors under a suitable proportional-marginal assumption.