Universal upper bound conjecture for slack matrices of polygons

Let $S_n$ be the slack matrix of any $n$-gon, and let $\lfloor x\rfloor$ denote the largest integer not exceeding $x$. Polygon slack-matrix upper-bound conjecture.

$$\operatorname{rank}_+(S_n)\leq\left\lfloor\frac{n+6}{2}\right\rfloor,$$

and equality holds for $5\leq n\leq15$. The conjecture is motivated by computational experiments on generic polygons; the source reports no proof or disproof.