Converse characterization of groups by directed endomorphism graphs

Prove that if two groups have isomorphic directed endomorphism graphs, then the groups themselves are isomorphic.

Background

The paper establishes that isomorphic groups have isomorphic directed endomorphism graphs, using a group isomorphism to transport endomorphisms and hence directed edges between the graphs. It also shows that the analogous converse fails for undirected endomorphism graphs: the non-isomorphic groups $\mathbb{Z}_4$ and the Klein four group both have complete endomorphism graphs $K_4$.

The explicitly stated conjecture asks whether this loss of information is avoided in the directed setting. Namely, it proposes that the directed endomorphism graph determines the group up to isomorphism, so that isomorphic directed endomorphism graphs would imply isomorphic underlying groups.