Distortion conjecture for the variant correspondence induced by $\alpha$

Let $\alpha:\mathbb{S}^1\to\mathbb{S}^2$ be the embedding

$$\alpha(t)=\big(\cos(t)\sqrt{1-z^2(t)},\sin(t)\sqrt{1-z^2(t)},z(t)\big),\qquad z(t)=0.15\,\cos(3t),$$

and let $R_\alpha$ be its induced correspondence. Variant correspondence conjecture.

$$\mathrm{dis}(R_\alpha)=\frac{2\pi}{3}.$$

This correspondence arose from computational experimentation as a variant of $R_2$; the claimed optimality remains unproved in the supplied text.