Madiman–Kontoyannis conjecture for entropy of differences of log-concave random vectors

Let $X$ and $Y$ be iid $\mathbb{R}^d$-valued random variables with log-concave distributions and differential entropy $h$. Their difference $X-Y$ is also $\mathbb{R}^d$-valued.

Madiman–Kontoyannis conjecture. The entropy satisfies

$$h(X-Y)\leq h(X)+d\log 2,$$

with equality when $X$ and $Y$ have the $d$-dimensional exponential product distribution.

This is the entropic analogue of the Rogers–Shephard inequality for convex bodies. The claim is attributed in the source to Madiman and Kontoyannis; the source provides no resolution status.