Gromov–Hausdorff distance conjecture for a regular polygon and the circle

Let $S^1$ be the circle of unit radius in $\mathbb{R}^2$ centered at the origin, and let $P_n$ be an inscribed regular polygon with $n$ sides; equip both with the Euclidean metric. Let $d_{GH}$ denote Gromov–Hausdorff distance and $d_H^{\mathbb{R}^2}$ Hausdorff distance in the common ambient space $\mathbb{R}^2$. Gromov–Hausdorff distance conjecture.

$$d_{GH}(P_n,S^1)=d_H^{\mathbb{R}^2}(P_n,S^1)=1-\cos\left(\frac{\pi}{n}\right).$$

The Hausdorff distance gives the displayed upper bound for the Gromov–Hausdorff distance via the common Euclidean embedding. The conjecture asserts that this upper bound is sharp; no resolution is supplied in the given text.