Alexandersson–Amini's cyclic sieving conjecture for stretched skew tableaux

Let $\lambda/\mu$ be a skew shape, let $m$ be a positive integer, and let $k$ be a positive integer. Write $m\lambda/m\mu$ for the skew shape obtained by multiplying every part of $\lambda$ and $\mu$ by $m$, and let $\operatorname{SSYT}(m\lambda/m\mu,k)$ denote the set of semistandard Young tableaux of this shape with entries in $\{1,\ldots,k\}$. Let $s_{m\lambda/m\mu}(1,q,q^2,\dotsc,q^{k-1})$ be the corresponding specialized skew Schur polynomial. Alexandersson–Amini's conjecture. There is an action $\beta$ on $\operatorname{SSYT}(m\lambda/m\mu,k)$ of order $m$ such that

$$\left(\operatorname{SSYT}(m\lambda/m\mu,k),\langle\beta\rangle,s_{m\lambda/m\mu}(1,q,q^2,\dotsc,q^{k-1})\right)$$

exhibits the cyclic sieving phenomenon. This conjecture proposes a cyclic action realizing the principal specialization of the stretched skew Schur polynomial as a cyclic sieving polynomial; the source presents it as an open question, and no resolution is supplied here.