Completely graded decomposition conjecture for semisimple complex algebras

Let $A$ be a finite-dimensional semisimple complex algebra of the form specified in the source, with multiplicities $b(n)$ in its decomposition. A connected grading of $A$ is completely decomposed if all its graded-simple summands are simple, and $A$ is completely graded decomposition (CGD) if all its connected gradings are completely decomposed. Completely graded decomposition conjecture. The algebra $A$ is CGD if all the multiplicities $b(n)$ are at most $1$. The preceding proposition proves the necessity of the condition; the converse is stated as likely to hold and remains open in the supplied text.