Existence of essential cubical models without separating simplices

Investigate whether every one-ended cubulable group G can be realized as π1(X) for some essential compact non-positively curved cube complex X such that no vertex link in X contains a separating simplex.

Background

The paper provides a geometric method (via unfolding in square complexes and Shepherd’s theorem in higher dimensions) to compute Grushko decompositions and to avoid 1-cuts when π1(X) is one-ended.

Recognizing essentiality and avoiding separating simplices in links are important for algorithmically deciding free splittings; this question asks if such a favorable cubical model exists for all one-ended cubulable groups.