Eventual evenness of standard Young tableau counts for generic partitions

Let $c$ be a constant. A partition $\lambda$ of $n$ boxes is called generic if its Young diagram fits into a $c\sqrt n$ by $c\sqrt n$ grid. Write $\operatorname{SYT}(\lambda)$ for the set of standard Young tableaux of shape $\lambda$. Eventual evenness conjecture. For every constant $c$, there is a number of boxes $b$ such that, for all $n>b$, every generic partition $\lambda$ of $n$ boxes has an even number of standard Young tableaux:

$$|\operatorname{SYT}(\lambda)|\equiv 0 \pmod{2}.$$

The conjecture is motivated by the hook-length formula and would explain why the cactus-group image is eventually more often contained in an alternating group for generic partitions; its status is not resolved in the source.