Berge–Füredi–Meyniel conjecture on the chromatic index of linear hypergraphs

For a hypergraph $H$ and a vertex $v\in V(H)$, define its neighborhood by

$$N(v):=\bigcup_{v\in e\in E(H)}(e\setminus\{v\}).$$

Let $\Delta^*(H)$ be the maximum of $|N(v)|$ over all vertices $v\in V(H)$. A linear hypergraph is one in which any two distinct edges intersect in at most one vertex, and its chromatic index is the minimum number of colors needed to color its edges so that intersecting edges receive different colors.

Berge–Füredi–Meyniel conjecture. For any linear hypergraph $H$, its chromatic index is at most $\Delta^*(H)+1$.

This is presented as a generalization of the Erdős–Faber–Lovász conjecture. The supplied text describes it as a conjectural strengthening and does not state that it has been resolved.