Cyclotomic nondivisibility 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 nondivisibility conjecture. For every $n\geq 2$ and every integer $i$ satisfying $2^i\leq n$,

$$1+x^{2^i}\nmid \mathrm{num}_{\mathcal B}(n,x).$$

The paper gives evidence for this conjecture, which remains open.