Eventual alternating-group containment for cactus-group images

Let $C_n$ be the cactus group acting on the set of standard Young tableaux of a partition, and let $A_N$ denote the alternating group on $N$ tableaux, where $N$ is the number of standard Young tableaux. A partition is generic when its Young diagram fits into the prescribed square grid. Alternating-group containment conjecture. There exists some $m$ such that, for all generic partitions with $n>m$ boxes, the image of $C_n$ is contained in $A_N$:

$$\operatorname{im}(C_n)\subseteq A_N.$$

This is presented as a consequence of the preceding eventual-evenness conjecture and concerns the eventual parity of the cactus-group action on standard Young tableaux. The source gives no resolution, so the claim remains open.