Goulden–Jackson's $b$-conjecture for hypermap coefficients

Let $\tau$, $\mu$, and $\nu$ be partitions of $n\geq 1$. The coefficient $h_{\mu,\nu}^{\tau}(\beta)$ is defined through the power-sum expansion of the connected Jack three-point series, and a rooted bipartite map has face distribution $\tau$, black vertex distribution $\mu$, and white vertex distribution $\nu$. $b$-conjecture. For all partitions $\tau,\mu,\nu\vdash n\geq 1$, the quantity $h_{\mu,\nu}^{\tau}(\beta)$ is a polynomial in $\beta$ with nonnegative integer coefficients. Moreover, there exists a statistic $\eta$ on maps such that

$$h_{\mu,\nu}^{\tau}(\beta)=\sum_{\mathcal{M}}\beta^{\eta(\mathcal{M})},$$

where the sum runs over all rooted bipartite maps $\mathcal{M}$ with face distribution $\tau$, black vertex distribution $\mu$, and white vertex distribution $\nu$, and $\eta(\mathcal{M})$ is a nonnegative integer equal to $0$ if and only if $\mathcal{M}$ is orientable. This is the conjecture of Goulden and Jackson motivating the shifted parameter $\beta=\alpha-1$; it remains open.