Entropy concavity for log-concave random variables: an asymmetric counterexample

The Ball-Nayar-Tkocz entropy concavity conjecture asserts that, if $X,Y$ are independent identically distributed real random variables with a common log-concave density, then the differential entropy of their weighted sum, $$F(t)=h(\sqrt{1-t}X+\sqrt{t}Y),0\le t\le 1,$$ is a concave function of the weight parameter $t$. We construct an asymmetric, strictly positive smooth probability density $f$ with mean zero, variance one, and $(\log f{)}^{\mathrm{\prime }\mathrm{\prime }}<-1\mathrm{/}2$, for which the corresponding function satisfies $F^{\mathrm{\prime }\mathrm{\prime }}(t)>0$ throughout an endpoint neighborhood $0<t<\delta$. This disproves the conjecture without an additional symmetry assumption. For a class of Gaussian perturbation densities, we first establish the endpoint expansion $$F^{\mathrm{\prime }\mathrm{\prime }}(t)=-\frac{{\mu }_3{J}_3}{16\sqrt{t}}+O(1),t\downarrow 0.$$ Here ${\mu }_3=\int x^{3}f(x)dx$ is the third central moment. Writing $\rho =(\log f{)}^{\mathrm{\prime }}$ for the score function, its third moment is $J_3=\int f(x)\rho (x{)}^{3}dx$. A Hermite perturbation gives ${\mu }_3>0$ and $J_3<0$, and explicit remainder estimates verify the constructed density and its endpoint curvature. The counterexample does not address the conjecture with an additional symmetry assumption.