Periodic 2-cuts characterizing Z-splittings without hyperbolicity

Determine whether every compact, essential, non-positively curved cube complex X without 0- or 1-cuts whose fundamental group π1(X) splits over Z necessarily admits a periodic 2-cut.

Background

Theorem 3.5 establishes that in the hyperbolic setting, π1(X) splits over Z if and only if X admits a periodic 2-cut, using the algebraic annulus theorem and Whitehead complexes.

Extending this characterization beyond hyperbolic cube complexes is open; one direction (periodic 2-cut implies a Z-splitting) follows from the algebraic annulus theorem, but necessity remains unproven.