3-geodesic conjecture for Dantzig figures

A $d$-dimensional Dantzig figure is denoted by $(P,x,y)$, where $P$ is its graph and $x,y$ are the distinguished vertices. Let $m(P:x,y)$ be the maximum number of point-distinct geodesics between $x$ and $y$, where point-distinct means that any two geodesics have only $x$ and $y$ in common.

3-geodesic conjecture. For every $d$-dimensional Dantzig figure $(P,x,y)$,

$$m(P:x,y)\geq 3.$$

The paper states that this condition would imply the Strong Dantzig conjecture in that dimension, and hence would provide an approach to the Hirsch conjecture. No resolution is supplied in the source, so the conjecture remains open.