Cyclic sieving conjecture for staircase flagged tableaux

Let $sc_n=(n,n-1,\ldots,2,1)$ and let $b=(\ell+1,\ell+2,\ldots,\ell+n)$. Define

$$Cat_{\ell}(x):=\prod_{j=0}^{\ell-1}\prod_{i=1}^{n}\frac{1-x^{n+1+i+2j}}{1-x^{i+2j}}.$$

Staircase flagged-tableaux cyclic sieving conjecture. The triple $(\mathrm{FT}(sc_n,b),\langle\operatorname*{Pro}\rangle,Cat_{\ell}(x))$ exhibits the cyclic sieving phenomenon.

This conjecture is stated in terms of flagged tableaux by Serrano and Stump and in terms of multi-cluster complexes by Ceballos, Labbé, and Stump. The case $\ell=1$ follows from Eu and Fu, but the case $\ell>1$ remains open.