The hypercellular complex characterization conjecture

Let $G$ be a graph. A hyperprism complex is a polyhedral cell complex whose cells are Cartesian products of segments and regular polygons with an even number of sides, glued so that the intersection of any two cells is a cell; its 1-skeleton is the graph formed by its 0-cells and 1-cells. A hypercellular complex is a cell complex of a hypercellular graph. A polygonal complex is a complex whose 2-cells are regular polygons with an even number of sides. The 3C-condition and 3CC-condition are the corresponding conditions described in the paper.

The hypercellular complex characterization conjecture. For a graph $G$, the following conditions are equivalent:

$$\begin{array}{ll} \text{(i)} & G \text{ is hypercellular};\\ \text{(ii)} & G \text{ is the 1-skeleton of a simply connected hyperprism complex }{\bf X}\text{ satisfying the 3C-condition};\\ \text{(iii)} & G \text{ is the 1-skeleton of a simply connected polygonal complex }{\bf X}\text{ whose 2-cells are regular polygons with an even number of sides and which satisfies the 3CC-condition}. \end{array}$$

Moreover, all hypercellular cell complexes are CAT(0) spaces. This would provide a geometric characterization extending the analogous characterization of median graphs by simply connected cube complexes satisfying the cube condition. It would also imply the fixed point property used earlier in the paper, but the conjecture is not resolved in the supplied text.