Folklore conjecture on the reverse entropy power inequality for i.i.d. log-concave random vectors

Let $X$ and $Y$ be independent and identically distributed log-concave random vectors taking values in $\mathbb{R}^d$, and let $h$ denote differential entropy. Folklore conjecture. The entropy increment $h(X+Y)-h(X)$ is maximized when $X$ and $Y$ have exponential distributions. This conjecture seeks the sharp reverse entropy power inequality in the i.i.d. log-concave setting, where no linear volume-preserving repositioning is needed. Its status is not specified in the source.