Strong Dantzig conjecture for fundamental deformations

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.

Strong Dantzig conjecture. Every fundamental deformation of a $d$-dimensional Dantzig figure is good; equivalently,

$$E(d)=D(d).$$

The source explains that $D(d)$ is connected in the strong sense, so this conjecture would imply the Dantzig conjecture. It is presented as an open conjecture within the proposed approach to the Hirsch conjecture.