Reconstruction conjecture for cubical polytopes

A cubical polytope is a polytope whose facets are cubes. Its dual graph is the graph whose vertices correspond to facets of the polytope, with two vertices adjacent when the corresponding facets share a ridge.

Cubical-polytope reconstruction conjecture. Every cubical polytope can be reconstructed from its dual graph.

This conjecture asks whether the dual graph determines the entire combinatorial structure of a cubical polytope, paralleling the reconstruction result for simple polytopes from their graphs. The source presents it as a tempting conjecture and gives no resolution.