The injective map expansion conjecture for Jack polynomials

Let $\lambda$ be a partition of size $n$, and let $\operatorname{\mathcal{ID}}_n(\lambda)$ denote the set of injectively decorated maps associated with $\lambda$. Write $\operatorname{\mathcal{V}_\circ}(M)$ for the relevant vertex set, $\operatorname{cc}(M)$ for the number of connected components, and $\operatorname{face-type}(M)$ for the face type of $M$. Let $b=\alpha-1$.

Injective map expansion conjecture. There exists a statistic of non-orientability $\vartheta$ on $\operatorname{\mathcal{ID}}_n(\lambda)$ such that

$$J_\lambda^{(\alpha)}(\mathbf{p})=\sum_{M\in\operatorname{\mathcal{ID}}_n(\lambda)}(-1)^{n-|\operatorname{\mathcal{V}_\circ}(M)|}\alpha^{|\operatorname{\mathcal{V}_\circ}(M)|-\operatorname{cc}(M)}b^{\vartheta(M)}p_{\operatorname{face-type}(M)}.$$

This conjecture seeks an injective, map-theoretic refinement of the Jack-polynomial expansion; the paper proves the conjecture for the coefficients of $\alpha^0$ and $\alpha^1$ for every partition, and for all two-column partitions, while the general case remains open.