Entropy concavity for symmetric log-concave densities

Determine whether the function h(√(1−t)X+√tY) is concave in t∈[0,1] when X and Y are independent identically distributed real random variables whose common log-concave density is symmetric about the origin.

Background

The paper disproves the Ball–Nayar–Tkocz entropy concavity conjecture for general log-concave densities by constructing an asymmetric, strictly positive, smooth, strongly log-concave density for which the entropy of the weighted sum is strictly convex near the endpoints. The counterexample relies on nonzero third moments of both the density and its score function.

For a density symmetric about the origin, the third central moment and the third score moment both vanish, so the singular t^{-1/2} curvature term used to produce the asymmetric counterexample also vanishes. Consequently, the paper does not resolve whether entropy concavity holds under the additional symmetry assumption.