Ballantine–Beck–Feigon–Maurischat's coprimality conjecture for partition polynomials

Let $\operatorname{num}(n,x)$ and $\operatorname{den}(n,x)$ be the ordinary numerator and denominator of the reciprocal of the partition polynomial for $n\geq 1$. Ballantine–Beck–Feigon–Maurischat's coprimality conjecture. For every $n\geq 1$,

$$\gcd(\operatorname{num}(n,x),\operatorname{den}(n,x))=1.$$

This conjecture asks whether the displayed numerator and denominator are already in lowest terms; its resolution is not specified in the supplied text.