The measure-of-non-orientability conjecture for Jack characters

Let $\lambda$ be a Young diagram, let $\pi$ be a partition, let $\alpha>0$, and let $M$ range over non-oriented maps with face-type $\pi$. Write $V_\bullet(G)$ and $V_\circ(G)$ for the two vertex classes of the underlying bipartite graph $G$, and let $N_M(\lambda)$ be the associated map function. Let $\operatorname{Ch}_\pi^{(\alpha)}(\lambda)$ denote the Jack character and let $\operatorname{mon}_M$ be the measure of non-orientability defined in the source. The measure-of-non-orientability conjecture.

$$\operatorname{Ch}_\pi^{(\alpha)}(\lambda)=(-1)^{\ell(\pi)}\sum_M\left(-\frac{1}{\sqrt{\alpha}}\right)^{|V_\bullet(G)|}\left(\sqrt{\alpha}\right)^{|V_\circ(G)|}\operatorname{mon}_M N_M(\lambda).$$

This conjecture gives a concrete interpretation of the coefficients in the Jack-character map expansion as measures of non-orientability; it is known for rectangular Young diagrams, while the general assertion is not established.