Ballantine–Beck–Feigon–Maurischat's unimodality conjecture for binary partition numerators

For binary partitions, let $\operatorname{num}_{\mathcal B}(n,x)$ denote the binary numerator. Ballantine–Beck–Feigon–Maurischat's binary unimodality conjecture. For every $n>5$, the polynomial $\operatorname{num}_{\mathcal B}(n,x)$ is unimodal. The supplied text explains that an earlier log-concavity formulation fails in small cases and that the intended log-concavity version excludes $n=4,5$ and remains open for $n>5$; the stated unimodality conjecture itself is not resolved in the supplied text.