The logarithmic psd-rank conjecture for regular polygon slack matrices

Let $S_n$ be the slack matrix of the regular $n$-gon, and let $\operatorname{rank}_{\operatorname{psd}}(S_n)$ denote its positive semidefinite rank. Logarithmic psd-rank conjecture. The psd-rank of $S_n$ is given by

$$\operatorname{rank}_{\operatorname{psd}}(S_n)=1+\lceil\log_2(n)\rceil.$$

This conjecture proposes an exact logarithmic formula for the positive semidefinite rank of regular polygon slack matrices. The preceding computations support the formula for the smallest regular polygons and suggest it for $n\geq 7$, but the statement is not established in the provided text.