Conformal-block expansion conjecture for the CUE moment function

Let $f_k(x)$ be the function defined in equation (a5), let $N$ be the parameter appearing there, and let $\mathcal{B}_{\lambda,\mu}$ denote the conformal blocks defined in equation (CBPV2), indexed by Young diagrams $\lambda,\mu\in\mathbb{Y}$. Let $\widetilde{\mathcal C}_0$ be the coefficient defined in equation (Ctilde). Conformal-block expansion conjecture. The function $f_k(x)$ has the following conformal block expansion near $x=0$:

$$f_k(x)=\frac{e^{-\frac{N}{2}x-kx}}{G(N+k+1)^2}\sum_{n=0}^{N}\widetilde{\mathcal C}_0\left(\frac{N+k}{2};k;k+n\right)x^{2nk+n^2}\sum_{\lambda,\mu\in\mathbb{Y}}\mathcal{B}_{\lambda,\mu}\left(\frac{k+N}{2},\frac{k+N}{2},k,k+n\right)x^{|\lambda|+|\mu|}.$$

Here the coefficients and conformal blocks are those specified in the cited definitions. The formula is conjectural because it is proposed after matching the Painlevé expansion with the CUE moment function; establishing the expansion would provide an explicit finite conformal-block representation of $f_k(x)$ near the origin.