Full-range codimension conjecture for homomesic spaces of fences

Let $F$ be a fence with $t$ segments. Let $I_H(F)$ be its ideal-statistic homomesic space and $I_T(F)$ its ideal-toggleability space. Full-range codimension conjecture. For every integer $t\geq 2$ and every integer $i$ with $0\leq i\leq t-1$, there exists a fence $F$ with $t$ segments such that

$$\dim(I_H(F))-\dim(I_T(F))=i.$$

The claim is based on computer experiments for fences with at most seven segments. The paper proves the corresponding toggleability-space dimension formula, but does not prove this full-range assertion.