The one-generation conjecture for the block algebra

Let $\SB$ be an exceptional block, and let $A_\SB$ be the associated $A_\infty$ algebra. One-generation conjecture. The algebra $A_\SB$ is one-generated as an $A_\infty$ algebra, and its Koszul dual is an ordinary algebra. The conjecture is motivated by the explicit example in the paper; the general assertion is left open.