The total coloring conjecture in fractional-power form

Let $G$ be a simple graph. The graph $G^{\frac{2}{2}}$ is the square of the subdivision of $G$ obtained by replacing every edge by a path of length two, and $\chi$ and $\omega$ denote its chromatic and clique numbers. Total coloring conjecture. For every simple graph $G$,

$$\chi(G^{\frac{2}{2}})\leq\omega(G^{\frac{2}{2}})+1.$$

Since $\omega(G^{\frac{2}{2}})=\Delta(G)+1$, this is the fractional-power reformulation of the total coloring conjecture. The source reports no resolution status for this formulation.