Tenner's iterative Glaisher bijection conjecture for noncoprime regular and distinct partitions

Let $s$ and $t$ be positive integers that are not coprime, and let $\phi_m$ denote Glaisher's involution with modulus $m$. An $s$-regular partition has no parts divisible by $s$, and a $t$-distinct partition has no part appearing $t$ or more times. Likewise, define $s$-distinct and $t$-regular partitions analogously.

Tenner's conjecture. Iteration of the map $\phi_s\phi_t$ suffices to produce a bijection: for every $s$-regular, $t$-distinct partition $\lambda$, there exists an integer $\ell$ depending on $\lambda$ such that $(\phi_s\phi_t)^\ell(\lambda)$ is the unique $s$-distinct, $t$-regular partition in its orbit, and no intervening $(\phi_s\phi_t)^k(\lambda)$ is $s$-regular and $t$-distinct.

For coprime $s$ and $t$, a single application of the preceding map gives the corresponding bijection; this conjecture proposes that repeated application works when $s$ and $t$ are not coprime.