Unimodality and log-concavity conjecture for binary-partition numerators

Let $\mathcal B(n)$ be the set of binary partitions of $n$, and let $\mathrm{num}_{\mathcal B}(n,x)$ be the associated numerator polynomial. Binary-partition coefficient conjecture. For every $n\geq 2$, $\mathrm{num}_{\mathcal B}(n,x)$ is unimodal and log-concave. The claim is supported by the discussion of coefficient behavior and remains open.