Hyperfocal subalgebra stable unital basis conjecture

Let $b$ be a block of $\mathcal{O}G$ with defect group $D$, let $A=l\mathcal{O}Gl$ be a source algebra, and let $\mathcal{F}$ be the saturated fusion system of $A$ on $D$. Write $\mathrm{hyp}(\mathcal{F})$ for its hyperfocal subgroup. For a normal subgroup $\tilde{D}\trianglelefteq D$ containing $\mathrm{hyp}(\mathcal{F})$, let $\tilde{A}$ be the hyperfocal subalgebra with respect to $\tilde{D}$, set $\bar{D}:=D/\tilde{D}$, and let $D\times^{\bar{D}}D$ be the corresponding subgroup of $D\times D$. Hyperfocal subalgebra conjecture. For any source algebra $A=l\mathcal{O}Gl$ of any block $b$ with defect group $D$, there is a normal subgroup $\tilde{D}$ in $D$ containing $\mathrm{hyp}(\mathcal{F})$ such that the hyperfocal subalgebra $\tilde{A}$ with respect to $\tilde{D}$ has a $D\times^{\bar{D}}D$-stable unital basis. The paper introduces this conjecture as a reduction of the Barker–Gelvin stable unital basis conjecture; the supplied text gives no evidence that it has been resolved.