The circular altitude conjecture for the third Mycielskian of $K_6$

Let $K_6$ be the complete graph on six vertices, and let $\mathcal M^3(K_6)$ denote its third Mycielskian. Write $\alpha^\circ(G)$ for the circular altitude of a graph $G$ and $\chi_c(G)$ for its circular chromatic number. The preceding bounds give

$$8\leqslant \alpha^\circ(\mathcal M^3(K_6))\leqslant \chi_c(\mathcal M^3(K_6))\leqslant 9.$$

Circular altitude conjecture.

$$\alpha^\circ(\mathcal M^3(K_6))=9$$

and hence

$$\chi_c(\mathcal M^3(K_6))=9.$$

The circular chromatic number of $\mathcal M^3(K_6)$ is a prominent open problem in circular colouring. The conjecture is based on trial computer computations and would determine both the circular altitude and circular chromatic number in this case.