Homogeneous boolean rank conjecture for polygon slack matrices

Let $S_n$ be the slack matrix of an $n$-gon. Its homogeneous boolean rank is denoted by ${\textup{rank}\,^{\textup{hom}}_{\mathbb{B}}}(S_n)$, while ${\textup{rank}\,_{\mathbb{B}}}(S_n)$ denotes its boolean rank.

Homogeneous boolean rank conjecture. For any $n$,

$${\textup{rank}\,^{\textup{hom}}_{\mathbb{B}}}(S_n)={\textup{rank}\,_{\mathbb{B}}}(S_n).$$

The homogeneous boolean rank restricts boolean factorizations by requiring fixed support sizes in the factor matrices. The paper notes that the homogeneous rank is at least the usual boolean rank and reports agreement in several computed ranges, but the equality is not established for every $n$.