Second-term oscillation conjecture for hook Young-diagram distribution functions

Let $k, ell\ge 0$ and $\beta,z>0$. Define

$$s_{(k,\ell),\beta}(n,z)=\frac{\sum_{\lambda\in H(k,\ell;n,z)}(f^\lambda)^\beta}{\sum_{\lambda\in H(k,\ell;n)}(f^\lambda)^\beta}$$

and let $r_{(k,\ell),\beta}(z)$ be the corresponding limiting integral ratio. Second-term oscillation conjecture. As $n\to\infty$, the quantity

$$\frac{\sqrt n}{k+\ell}\left[s_{(k,\ell),\beta}(n,z)-r_{(k,\ell),\beta}(z)\right]$$

oscillates in a symmetric bounded interval $(-L(k,\ell,\beta,z),L(k,\ell,\beta,z))$ centered at zero; its endpoints are conjectured to be the infimum and supremum of the values of this quantity. This is a numerical, explicitly described conjecture about the second-order approximation to the distribution function; the source provides no resolution.