Barker–Gelvin stable unital basis conjecture for source algebras

Let $\mathcal{O}$ be a complete local noetherian commutative ring with algebraically closed residue field of prime characteristic $p$. Let $G$ be a finite group with $p\mid |G|$, let $b$ be a block of $\mathcal{O}G$ with defect group $D$, and let $A=l\mathcal{O}Gl$ be a source algebra of $\mathcal{O}Gb$, for a suitable primitive idempotent $l\in(\mathcal{O}Gb)^D$. A basis is unital if it contains a unit, and $D\times D$-stable if it is preserved by the left and right action of $D$. Barker–Gelvin's conjecture. For any block $b$ of $\mathcal{O}G$, any source algebra $A$ of $\mathcal{O}Gb$ has a $D\times D$-stable unital basis. This conjecture concerns the existence of particularly well-behaved bases for source algebras and is the conjecture investigated in the paper; the supplied text does not state whether it has been resolved.