Erdős–Nešetřil conjecture on the strong chromatic index

Let $G$ be a finite simple graph with maximum degree $\Delta$.

Erdős–Nešetřil conjecture. The strong chromatic index of $G$ satisfies

$$\chi'_s(G)\le\begin{cases} \frac{5}{4}\Delta^2 &\text{if } \Delta \text{ is even}, \\ \frac{5}{4}\Delta^2-\frac{1}{2}\Delta+\frac{1}{4} &\text{if } \Delta \text{ is odd}. \end{cases}$$

The first non-trivial case, $\Delta=3$, has been verified, but the conjecture remains open for $\Delta\ge 4$.