The shrub-depth characterization by vertex-minors

Let $\mathcal{C}$ be a class of graphs. A graph is of bounded shrub-depth if its shrub-depth is bounded by some fixed integer.

Shrub-depth characterization conjecture. The class $\mathcal{C}$ is of bounded shrub-depth if, and only if, there exists an integer $t$ such that no graph $G\in\mathcal{C}$ contains a path of length $t$ as a vertex-minor.

This asks whether excluding a fixed path as a vertex-minor exactly characterizes classes of graphs of bounded shrub-depth. The surrounding discussion presents it as a natural question following the construction of cographs containing arbitrarily long paths as pivot-minors; its resolution is not specified in the source.