Strong Coloring Conjecture for the strong chromatic number

Let $H$ be a graph, let $\Delta(H)$ denote its maximum degree, and let $schi(H)$ denote its strong chromatic number, the minimum number of colors needed so that every graph formed by adding a clique to the neighborhood of each vertex is properly colorable with those colors. Strong Coloring Conjecture. For any graph $H$,

$$s\chi(H)\leq 2\Delta(H).$$

The bound would be best possible, since for each maximum degree $\Delta$ there are $\Delta$-regular graphs with strong chromatic number at least $2\Delta$. The conjecture remains open for all $\Delta\geq 2$, although several partial results are known for general $\Delta$.