The retract characterization conjecture for hypercellular graphs

Let $G$ be a partial cube. A retract of a graph $H$ is a subgraph admitting a graph homomorphism from $H$ to the subgraph that restricts to the identity on the subgraph. A Cartesian product of graphs is formed by taking vertex tuples and joining tuples that differ in exactly one coordinate along an edge in that factor. Bipartite cellular graphs are the class used in the paper.

The retract characterization conjecture. A partial cube $G$ is hypercellular if and only if $G$ is a retract of a Cartesian product of bipartite cellular graphs.

This is proposed as an analogue of the fact that median graphs are precisely retracts of hypercubes. The supplied text gives no resolution, so the conjecture remains open.