Extremal monotonicity conjecture for the side length function of regular polygon stars

Let $P_n$ be a regular polygon with $n$ sides, let $s_{2l+1}$ denote the side length function for inscribed $(2l+1)$-pointed stars, and call a point of $P_n$ a midpoint crossing or vertex crossing according to whether the corresponding star has a midpoint coincidence or a vertex coincidence. For $n\ge 4l+2$, Extremal monotonicity conjecture. Every midpoint crossing of $P_n$ is a global minimum of $s_{2l+1}$, every vertex crossing is a global maximum, midpoint and vertex crossings are interleaved around $P_n$, and $s_{2l+1}$ is strictly monotonic between adjacent midpoint and vertex crossings. This would strengthen the preceding partial characterization, which establishes that crossings are local extrema and that all crossings of each type have a common value; the conjectured global ordering and strict monotonicity remain unproved in the supplied text.