On the Order of P-Strict Promotion on $V\times [\ell]$

Abstract: Denote by $V$ the poset consisting of the elements ${A,B,C}$ with cover relations ${A\lessdot B, A\lessdot C}$. We show that $P$-strict promotion, as defined by Bernstein, Striker, and Vorland, on $P$-strict labelings of $V\times [\ell]$ with labels in the set $[q]$ has order $2q$ for every $\ell\ge 1$ and $q\ge 3$ as conjectured by Bernstein, Striker, and Vorland. This resolves the equivalent conjecture of Hopkins that the order of piecewise-linear rowmotion on the order polytope of $V\times [k]$ has order $2(k+2)$ for all $k\ge 1$.