On the class-breadth conjecture

The class-breadth conjecture of Leedham-Green, Neumann and Wiegold states that for each $p$-group, $cl(G)\le b(G)+1$, where $cl(G)$, $b(G)$ denote the nilpotency class and the breadth of $G$. While several counter-examples to this conjecture have been found for $p=2$, it is still open in general for $p>2$. This article is dedicated to the general case $p>2$ of the conjecture.