Distortion conjecture for the tetrahedral spherical-harmonic correspondence

In spherical coordinates on $\mathbb{S}^2$, define

$$w(\phi,\theta)=\frac{1}{3}\sin(\theta)\sin(2\theta)\cos(2\phi)$$

and

$$\sigma(\phi,\theta)=\big(x(\phi,\theta)\sqrt{1-w^2(\phi,\theta)},y(\phi,\theta)\sqrt{1-w^2(\phi,\theta)},z(\phi,\theta)\sqrt{1-w^2(\phi,\theta)},w(\phi,\theta)\big),$$

where $(x(\phi,\theta),y(\phi,\theta),z(\phi,\theta))=(\cos(\phi)\sin(\theta),\sin(\phi)\sin(\theta),\cos(\theta))$. Let $R_\sigma$ be the induced correspondence and set $\zeta_2=\arccos(-1/3)$. Tetrahedral correspondence conjecture.

$$\mathrm{dis}(R_\sigma)=\zeta_2.$$

The text records the lower bound $\mathrm{dis}(R_\sigma)\geq\zeta_2$ and reports extensive computational testing, but does not establish the matching upper bound.