Ballantine–Beck–Feigon–Maurischat's ternary-partition evaluation conjecture

Let $\operatorname{num}_{\mathcal T}(n,x)$ denote the numerator associated with partitions into powers of $3$, and let $v_3(M)$ be the exponent of $3$ in the positive integer $M$. Define $s(n)\coloneq\operatorname{num}_{\mathcal T}(n,-1)$ for $n\geq 1$. Ballantine–Beck–Feigon–Maurischat's ternary-partition evaluation conjecture. We have $s(1)=s(2)=1$, and for every $n\geq 1$,

$$s(3n)=s(3n+1)=s(3n+2)=3^{v_3((3n)!)}.$$

This is a conjectured evaluation of ternary-partition numerators at $x=-1$; its resolution is not specified in the supplied text.