CUE joint-moment coefficient formula via Young diagrams

Let $F(h,k)$ be the large-$N$ limit

$$F(h,k):=\lim_{N\to\infty}\frac{1}{N^{k^2+2h}}F_N(h,k),$$

where $F_N(h,k)$ is the joint moment considered in the paper. For a Young diagram $\lambda$, write $|\lambda|$ for its size, $\lambda_1$ for its first row length, $\lambda'_j$ for the length of its $j$th column, and $h_\lambda(i,j)$ for the hook length of the box $(i,j)\in\lambda$. CUE moment coefficient conjecture. For $k>h-\frac12$,

$$F(h,k)=(-1)^h\frac{G(k+1)^2}{G(2k+1)}(2h)!\sum_{\substack{\lambda\in\mathbb{Y}\\|\lambda|=2h,\ \lambda_1\leq k}}\prod_{(i,j)\in\lambda}\frac{(2k+i-j)(k+i-j)}{h_\lambda^2(i,j)(\lambda'_j-i-j+1+2k)^2}.$$

This gives an explicit finite Young-diagram expression for the limiting joint moments and is motivated by the Painlevé III conformal-block expansion; proving it would establish the proposed formula for the coefficients $F(h,k)$ in the stated range.