Barker–Gelvin uniform source algebra conjecture for blocks

Let $G$ be a finite group, let $p$ divide $|G|$, and let $\bar{b}$ be a block idempotent of $kG$ with non-trivial defect group. A source algebra of $kG\bar{b}$ is the interior $D$-algebra associated with a source idempotent for $\bar{b}$, where $D$ is a defect group of $\bar{b}$. A source algebra is uniform if it admits a $D\times D$-stable unital basis. Barker–Gelvin's conjecture. Every source algebra of $kG\bar{b}$ is uniform. This conjecture asks whether source algebras of blocks over an algebraically closed field of prime characteristic always possess a basis compatible with the left and right defect-group actions and consisting of units. The supplied text gives no resolution, so the conjecture remains open.