Larson–Larson–Vogt intersection-class conjecture for transmission loci

Let $\cH_{g,k,2}$ denote the relevant Hurwitz space, and let $\cM_{g,2}$ denote the moduli space of twice-marked curves. For a transmission permutation $\tau$, let $W^\tau(C,p,q)$ be the associated transmission locus, let $\operatorname{inv}_k(\tau)$ be its expected codimension, let $N(\tau)$ be the number of reduced words for $\iota_{-\chi_\tau}\tau$, and let $\Theta$ denote the relevant theta class.

Larson–Larson–Vogt intersection-class conjecture. All statements of Larson–Larson–Vogt's Theorem 1.2 should generalize from splitting loci to transmission loci of a general point in $\cH_{g,k,2}$, or in $\cM_{g,2}$ when $k=0$. In particular,

$$[W^\tau(C,p,q)] = \frac{N(\tau)}{\operatorname{inv}_k(\tau)!}\Theta^{\operatorname{inv}_k(\tau)}.$$

Moreover, at any point of $\cH_{g,k,2}$, $W^\tau(C,p,q)$ should support this intersection class.

The source says that the formula is already known when $k=0$, while the case $k\geq 2$ is the primary open case and may be approachable by adapting the methods of Larson–Larson–Vogt.