Total geodesicity conjecture for simplicial embeddings of pants graphs

Let $\Sigma_{1}$ and $\Sigma_{2}$ be compact, orientable surfaces, and let $\mathcal{P}(\Sigma_i)$ denote the pants graph of $\Sigma_i$. A simplicial embedding is a map

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

that preserves adjacency and is injective. Total geodesicity conjecture. The image $\phi(\mathcal{P}(\Sigma_{1}))$ is totally geodesic in $\mathcal{P}(\Sigma_{2})$.

This would extend the known total-geodesicity results for subgraphs of pants graphs associated to suitable multicurves. The source presents it as a consequence anticipated from those results; no resolution is supplied here.