Preservation of log-concavity under Hadamard products

For a nonzero real polynomial $p$, let $\text{W}(p)$ denote the numerator of its ordinary generating function. We prove that if the coefficients of both $\text{W}(p)$ and $\text{W}(q)$ are nonnegative and log-concave with no internal zeros, then so are the coefficients of $\text{W}(pq)$. This provides an affirmative answer to a question of Brändén, Ferroni, and Jochemko. As applications, we derive corresponding results for finite products and for Cartesian products of lattice polytopes, answering a question of Ferroni and Higashitani.