Hopkins's rowmotion order conjecture for the V poset

Let $V=\{a,b,c\}$ be the three-element poset with $a\lessdot b$ and $a\lessdot c$, let $\mathcal{A}^{\ell}(P)$ denote the set of $P$-partitions of height $\ell$, and let $[m]=\{1,\ldots,m\}$. Hopkins's rowmotion order conjecture. Rowmotion on $\mathcal{A}^{\ell}(V\times[m])$ has order dividing

$$2(m+2).$$

The conjecture has been experimentally verified for small $m$ and $\ell$, with additional larger examples checked computationally, but no general proof is given.