The b-conjecture for Jack symmetric functions and nonorientable hypermaps

Let $\Psi(\mathbf{x},\mathbf{y},\mathbf{z};t,\alpha)$ be the Jack-symmetric-function generating series defined in the source, and let $b$ be an indeterminate with $\alpha=1+b$. A rooted hypermap is a map encoded by the corresponding coefficient of this series, with the orientable and all-surface cases obtained at $\alpha=1$ and $\alpha=2$, respectively.

The $b$-conjecture. The series $\Psi(\mathbf{x},\mathbf{y},\mathbf{z};t,1+b)$ has coefficients that are polynomials in $b$ with non-negative integer coefficients. The constant term at $b=0$ accounts for rooted hypermaps embedded in orientable surfaces, while the sum of all terms at $b=1$ accounts for rooted hypermaps embedded in all surfaces. Thus $b$ marks a statistic of nonorientability associated with rooted hypermaps.

This conjecture seeks a combinatorial interpretation of the Jack-parameter deformation between orientable and non-orientable map enumeration. The source reports it as conjectural, and no resolution is supplied here.