Blidia–Maffray–Zemir conjecture on the b-chromatic number of regular graphs

Let $G$ be a $d$-regular graph with girth at least $5$, and let $b(G)$ denote its b-chromatic number, the largest number of colors in a b-coloring of $G$. The Petersen graph is the 3-regular graph on 10 vertices with girth 5.

Blidia–Maffray–Zemir conjecture. If $G$ is not the Petersen graph, then

$$b(G)=d+1.$$

This predicts the maximum possible b-chromatic number for regular graphs of girth at least 5, apart from the Petersen graph, and remains open.