The gated-hull characterization conjecture for two-dimensional partial cubes

Let $G$ be a partial cube, and for each isometric cycle of $G$ consider its gated hull. A gated hull is a disk or a full subdivision according to the corresponding classes of cell complexes described in the paper.

Gated-hull characterization conjecture. If the gated hull of each isometric cycle of $G$ is a disk or a full subdivision, then $G$ is two-dimensional.

This conjecture asserts that condition (vii) in the characterization theorem is equivalent to conditions (i)--(vi), thereby giving a local characterization of two-dimensional partial cubes. The paper does not provide a proof or resolution of this converse implication.