Monotonicity conjecture for quantization coefficients of regular polygons

Let $m$ be a positive integer, and consider a regular $m$-sided polygon inscribed in a circle. The quantization coefficient is the coefficient associated with the asymptotic quantization error for the uniform distribution on the boundary of the polygon. For the unit circle, this coefficient is $\frac{\pi^2}{3}$, while for the boundary of a regular hexagon inscribed in a unit circle it is $3$.

Polygonal quantization conjecture. The quantization coefficient for the uniform distribution on the boundary of a regular $m$-sided polygon inscribed in a circle is an increasing function of $m$, and approaches the quantization coefficient for the uniform distribution on the circle as $m$ tends to infinity.

Regular polygons inscribed in a circle converge to the circle as their number of sides tends to infinity, so the conjecture predicts that the corresponding quantization coefficients increase toward the circular value. The supplied text gives the exact coefficients for the circle and the regular hexagon but provides no resolution of the general claim.