Witte–Forrester conjecture on Gaussian beta-ensemble one-point correlators

Let $W_1^{(l)}(x)$ be the one-point correlator at order $l$, let $\hbar$ be the expansion parameter, and write $y$ for the spectral-curve quantity appearing in these correlators. For each index $j$, let $P_j^l(x)$ be a polynomial in $x$.

Witte–Forrester conjecture. For even $l\leq 2$,

$$W_1^{(l)}(x)=\hbar^l\left[\frac{P_1^{l}(x)}{y^{3l-2}}+\frac{P_2^{l}(x)}{y^{3l-1}}\right]+\hbar^{l-2}\left[\frac{P_3^{l}(x)}{y^{3l-2}}+\frac{P_4^{l}(x)}{y^{3l-1}}\right]+\dots+\hbar^2 \left[\frac{P_{l-1}^{l}(x)}{y^{3l-2}}+\frac{P_l^{l}(x)}{y^{3l-1}}\right]+\frac{P_{l+1}^l(x)}{y^{3l-1}}.$$

Here $\operatorname{deg}_x P_j^l=l-1$ for $j=1,3,\dots,l-1$, $\operatorname{deg}_x P_j^l=l$ for $j=2,4,\dots,l$, and $\operatorname{deg}_x P_{l+1}^l=l-2$. For odd $l\geq 1$,

$$W_1^l(x)=\hbar^l\left[\frac{P_1^{l}(x)}{y^{3l-2}}+\frac{P_2^{l}(x)}{y^{3l-1}}\right]+\hbar^{l-2}\left[\frac{P_3^{l}(x)}{y^{3l-2}}+\frac{P_4^{l}(x)}{y^{3l-1}}\right]+\dots +\hbar \left[\frac{P_l^{l}(x)}{y^{3l-2}}+\frac{P_{l+1}^{l}(x)}{y^{3l-1}}\right].$$

In the odd case, $\operatorname{deg}_x P_j^l=l-1$ for $j=1,3,\dots,l$, while $\operatorname{deg}_x P_j^l=l$ for $j=2,4,\dots,l+1$. In both formulas, each polynomial is even or odd according to its degree, and the leading term of $W_1^l(x)$ as $x\to\infty$ has order $x^{-2l-1}$.