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

For ternary partitions, let $\operatorname{num}_{\mathcal T}(n,x)$ denote the associated numerator. Define $t(n)\coloneq\operatorname{num}_{\mathcal T}(n,1)$ for $n\geq 0$, with $t(0)\coloneq 1$. Ballantine–Beck–Feigon–Maurischat's ternary-partition recurrence conjecture. For every $n\geq 0$,

$$t(3n)=t(3n+1)=t(3n+2),$$

and for every $n\geq 1$,

$$t(n)=\frac{t(3n)-t(3n-2)}{2^{2n}}.$$

In particular, $t(1)=t(2)=1$. The supplied text explains that this is the intended correction of a literally false formulation in the original source; its resolution is not specified here.