Conjectured sharp Rogers-Shephard inequality for entropy

Let $X$ and $X'$ be independent random vectors in ${\mathbb R}^n$ drawn from the same log-concave distribution. Let $h$ denote differential entropy and define the entropy power by $\mathcal{N}(X)=\exp(2h(X)/n)$, with the associated difference constant satisfying $\mathcal{N}(X-X')\leq \sigma_{-}(X)^2\mathcal{N}(X)$. Conjectured sharp entropy Rogers-Shephard inequality. If $X$ and $X'$ are independent ${\mathbb R}$-valued random variables drawn from the same log-concave distribution, then

$$\mathcal{N}(X-X')\leq 4\mathcal{N}(X),$$

with equality if and only if $X$ is a translated and scaled version of the one-sided exponential distribution. Equivalently, for every log-concave random variable $X$, $\sigma_{-}(X)\leq 2$. This is proposed as the sharp one-dimensional analogue of the Rogers-Shephard inequality; the paper's preceding bound is $\sigma_{-}(X)\leq 8$, while contemporaneous work obtains a better non-sharp bound, so the optimal constant remains open.