Equivariant partite-presentation refinement for vertex transitive graphs

Let $\Gha$ be a vertex transitive graph with a partite presentation $P$ such that $\Gha=\operatorname{PCay}(P)$, and let $\Gpa\leq\operatorname{Aut}_{c\text{-}loc}(\Gha)$ be a transitive group. A cover is a graph map $\eta:\Gha\to\sC_{P'}$ compatible with the stated presentation structure. Equivariant presentation conjecture. There exists a partite presentation $P'$ such that $\Gha=\operatorname{PCay}(P')$, together with a cover $\eta:\Gha\to\sC_{P'}$ and a homomorphism $\phi:\Gpa\to\operatorname{Aut}(\sC_{P'})$ satisfying

$$\eta(g\cdot x)=\phi(g)\cdot\eta(x)$$

for $x\in V(\Gha)\cup\overrightarrow{E}(\Gha)$. Thus the action of $\Gpa$ commutes with the cover. The source presents this as a possible generalisation of the procedure, but does not explicitly label it as an open conjecture.