Ballantine–Beck–Feigon–Maurischat's binary numerator–denominator coprimality conjecture

For binary partitions, let $\operatorname{num}_{\mathcal B}(n,x)$ and $\operatorname{den}_{\mathcal B}(n,x)$ denote the corresponding numerator and denominator, where binary parts are powers of $2$. Ballantine–Beck–Feigon–Maurischat's binary coprimality conjecture. For every $n\geq 1$,

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

The supplied text states this conjecture is equivalent to the later binary divisibility conjecture; its resolution is not specified here.