The totally geodesic image conjecture for simplicial embeddings of pants graphs

Let $\Sigma_{1}$ and $\Sigma_{2}$ be compact, orientable surfaces, and let $\mathcal{P}(\Sigma)$ denote the pants graph of a surface $\Sigma$. A subgraph is totally geodesic if every geodesic in the ambient pants graph joining two of its vertices lies entirely in the subgraph. Totally geodesic image conjecture. If

$$\phi:\mathcal{P}(\Sigma_{1})\longrightarrow\mathcal{P}(\Sigma_{2})$$

is a simplicial embedding, then $\phi(\mathcal{P}(\Sigma_{1}))$ is totally geodesic in $\mathcal{P}(\Sigma_{2})$. This is posed as a guiding conjecture concerning the geometry of pants graphs; the supplied text does not state any resolution, so the conjecture remains open.