Pointwise convergence conjecture for hook-length mass functions

For $t \geq 4$, let $f_{t;n}(x)$ be the scaled mass function associated with the distribution of $\hat{Y}_t(n)$, and let $g_t(x)$ be the corresponding limiting function. Pointwise convergence conjecture. For all $x>0$,

$$f_{t;n}(x)\to g_t(x)\qquad\text{as }n\to\infty.$$

The conjecture asks whether the scaled probability mass functions converge pointwise in the regime $t\geq 4$. Convergence of the cumulative distribution functions to a shifted Gamma distribution is known, while the corresponding pointwise convergence of the mass functions had not been studied; numerical evidence supports the claim.