Relative codimension bound conjecture for transmission loci

Let $\pi:X\to B$ be a versal family in $\cM_{g,2}$ when $k=0$, or in $\cH_{g,k,2}$ when $k\geq 2$. Let $W^\tau(\pi,p,q)$ denote the corresponding relative transmission locus.

Relative codimension bound conjecture. Every component of $W^\tau(\pi,p,q)$ has codimension at most $\operatorname{inv}_k(\tau)$.

The source explains that this would yield a regeneration theorem: expected-dimensional transmission loci on singular curves would belong to components meeting nearby smooth curves with the expected dimension. The source does not report a proof or disproof.