Concavity conjecture for entropy along normalized sums of log-concave variables

Let $X$ and $Y$ be independent copies of a log-concave random variable, and let $\lambda\in[0,1]$. Define

$$f(\lambda)=\mathcal{S}(\sqrt{\lambda}X+\sqrt{1-\lambda}Y).$$

Concavity conjecture. The function $f$ is concave on $[0,1]$. This conjecture proposes a stronger form of regularity for entropy under normalized sums of independent identically distributed log-concave variables. The source gives no evidence of resolution.