Non-realizability of the triple (3,5,6) for graph parameters

Let $G$ be a graph, and write $\a0\omega(G)$ for its clique number, $\a0\chi(G)$ for its chromatic number, and $\a0\chi_{\rho}(G)$ for its packing chromatic number. Non-realizability conjecture. There is no graph $G$ such that

$$\omega(G)=3,\qquad \chi(G)=5,\qquad \chi_{\rho}(G)=6.$$

Equivalently, the triple $(3,5,6)$ is not realizable. The conjecture concerns whether the lower bound in the table of known values for the auxiliary function $m(a,b)$ can be improved; the supplied text gives no resolution, so its status remains open.