The sharp Gromov–Hausdorff distance conjecture for the circle

Let $Y$ be a geodesic, compact, and simply connected metric space, and let $d_{\mathrm{GH}}$ denote the Gromov–Hausdorff distance.

Circle distance conjecture. One has

$$d_{\mathrm{GH}}(\mathbb{S}^1,Y)\geq\frac{\pi}{3}.$$

The paper derives the weaker bound $d_{\mathrm{GH}}(\mathbb{S}^1,Y)\geq\frac{\pi}{6}$ by persistence-barcode methods and proposes the displayed inequality as a conjectural improvement. The source gives no resolution.