Coprimality conjecture for reciprocal subsum numerator and denominator

Let $n\geq 1$, and let $\mathrm{num}(n,x)$ and $\mathrm{den}(n,x)$ be the reduced numerator and denominator associated with the sum of reciprocals of subsum polynomials over partitions of $n$. Coprimality conjecture. For every $n\geq 1$,

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

The source describes this as a weaker claim for which partial results have been obtained; its general validity remains open.