Hanlon's permutation-pair expansion conjecture for Jack polynomials

Let $\lambda$ be a partition of size $n$, and let $\operatorname{\mathcal{ID}^O}_n(\lambda)$ denote the set of orientable injectively decorated maps associated with $\lambda$. Write $\operatorname{\mathcal{V}_\circ}(M)$ for the relevant vertex set and $\operatorname{face-type}(M)$ for the face type of $M$.

Hanlon's conjecture. There exists a statistic $w$ on the orientable maps in $\operatorname{\mathcal{ID}^O}_n(\lambda)$ such that

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

Hanlon formulated this conjecture as a weighted expansion of Jack polynomials in the power-sum basis using pairs of permutations. The claim is presented as an open reformulation, and the source gives no resolution.