The $b$-conjecture for unicellular bipartite maps

Let $n\geq 1$, and let $\tau$, $\mu$, and $\nu$ be partitions of $n$. For the coefficients $h_{\mu,\nu}^{\tau}(\beta)$ defined by the Jack-polynomial generating series, let the sum below range over rooted, bipartite maps $M$ whose face, black-vertex, and white-vertex distributions are respectively $\tau$, $\mu$, and $\nu$. $b$-conjecture. For every such $\tau$, $\mu$, and $\nu$, there is a nonnegative integer $\eta(M)$ for each map $M$ such that

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

Moreover, $\eta(M)=0$ if and only if $M$ is orientable. This conjecture seeks a combinatorial interpretation of the Jack-deformation coefficients, extending their known interpretations at $\beta=0$ and $\beta=1$ as counts of rooted orientable and general bipartite maps, respectively. The existence of the statistic $\eta(M)$ with the stated orientability criterion is the unresolved part.