The CMSO transduction characterization of bounded shrub-depth

Let $\cf G$ be a class of graphs, let ${\mathrm{CMSO}_1}\xspace$ denote counting monadic second-order logic with one free set-variable type, let $\tau$ be a ${\mathrm{CMSO}_1}\xspace$ transduction, and let $P_t$ denote the path on $t$ vertices. Transduction characterization conjecture. The class $\cf G$ has bounded shrub-depth if and only if, for every ${\mathrm{CMSO}_1}\xspace$ transduction $\tau$, there exists an integer $t$ such that

$$P_t\notin\tau(\cf G).$$

This conjecture proposes an asymptotic characterization of bounded shrub-depth through the transduction hierarchy: precisely the graph classes from which no counting monadic second-order transduction can produce paths of every length. The surrounding discussion presents it as plausible and does not give a resolution.