Ballantine–Beck–Feigon–Maurischat's unimodality conjecture for the even partition numerator

Write the ordinary partition numerator as $\operatorname{num}(n,x)=\operatorname{num}_0(n,x)+\operatorname{num}_1(n,x)$, where $\operatorname{num}_0(n,x)$ contains the even powers of $x$ and $\operatorname{num}_1(n,x)$ contains the odd powers. Ballantine–Beck–Feigon–Maurischat's unimodality conjecture. The polynomial $\operatorname{num}_0(n,x)$ is unimodal. This is a coefficient-shape conjecture for the ordinary numerator; its resolution is not specified in the supplied text.