Two-path obstruction characterization

Determine whether the forbidden-induced-subgraph characterization of bounded shrub-depth remains valid when the obstruction family of flipped $3P_t$ is replaced by flipped $2P_t$; equivalently, establish whether every hereditary graph class of unbounded shrub-depth contains arbitrarily large flipped $H_t$ or a flipped $2P_t$ as an induced subgraph.

Background

The paper characterizes hereditary graph classes of bounded shrub-depth by excluding, for some fixed order $t$, all flipped half-graphs of order $t$ and all flipped $tP_t$. It further observes that the multiplicity-$t$ path obstruction can be reduced to multiplicity three, yielding a characterization using flipped $3P_t$. The authors identify the multiplicity-two case as an unresolved strengthening, while multiplicity one cannot work because every graph on $t$ vertices is a flipped $1P_t$.