Coprimality conjecture for binary-partition reciprocal polynomials

Let $\mathcal B(n)$ be the set of binary partitions of $n$, and let $\mathrm{num}_{\mathcal B}(n,x)$ and $\mathrm{den}_{\mathcal B}(n,x)$ denote the associated numerator and denominator polynomials. Binary-partition coprimality conjecture. For every $n\geq 1$,

$$\gcd\bigl(\mathrm{num}_{\mathcal B}(n,x),\mathrm{den}_{\mathcal B}(n,x)\bigr)=1.$$

This is identified as the main conjecture for binary partitions and remains open.