Hecke-word Euler characteristic conjecture for transmission loci

Let $\widetilde{\Sigma}_k$ be the relevant set of transmission permutations, and let $\tau\in\widetilde{\Sigma}_k$ have shift $0$ and satisfy $\operatorname{inv}_k(\tau)\leq g$. Let $(C,p,q)$ be a general point of $\cM_{g,2}$ if $k=0$, or of $\cH_{g,k,2}$ if $k\geq 2$. A length-$g$ Hecke word for $\tau$ is a factorization

$$\tau=\alpha_1\star\alpha_2\star\cdots\star\alpha_g$$

where each $\alpha_i$ has shift $0$ and $\operatorname{inv}_k(\alpha_i)\leq 1$; equivalently, each $\alpha_i$ is either $\operatorname{id}$ or $\sigma^k_m$ for some $m\in\mathbb Z$. Let $H^\tau_g$ be the set of length-$g$ Hecke words for $\tau$.

Hecke-word Euler characteristic conjecture.

$$\chi\left(W^\tau(C,p,q),\mathcal O_{W^\tau(C,p,q)}\right)=(-1)^{g-\operatorname{inv}_k(\tau)}\#H^\tau_g.$$

The source motivates this as a generalization of the Euler-characteristic formula for Brill–Noether varieties and notes that better information on relative transmission loci, especially flatness criteria, is needed to prove it.