The finiteness conjecture for ghost numbers of group algebras

Let $G$ be a finite group, and let $k$ be a field whose characteristic divides the order of $G$. The ghost number finiteness conjecture. The ghost number of $kG$ is finite.

The conjecture extends the finiteness theorem proved when the principal block is generated by $k$, including the case where $G$ is a $p$-group. The general case remains open because the proof does not apply when the principal block is not generated by $k$.