Dantzig conjecture for the deformation graph of Dantzig figures

Let $D(d)$ be the graph of $d$-dimensional Dantzig figures under fundamental deformations, and let $E(d)$ be the subgraph with the same vertices whose edges correspond to good fundamental deformations. A directed path is an oriented path following the edge orientations.

Dantzig conjecture. The graph $E(d)$ is strongly connected: for any vertices $x,y$ of $E(d)$, there is an oriented path from $x$ to $y$.

This is introduced as one of the paper's basic conjectures in its proposed deformation program for approaching the Hirsch conjecture. The source does not give evidence of a resolution, so the conjecture remains open.