Existence conjecture for towers of algebras with prescribed Cartan invariants

Let $(A_n)_{n\geq 0}$ be a tower of algebras, and let $C^{(n)}$ denote the prescribed matrices associated with the tower's Cartan invariants. Existence conjecture. There exists a tower of algebras $(A_n)$ such that the $C^{(n)}$ are their matrices of Cartan invariants. In particular, one should have $\dim A_n=n!$. The conjecture is motivated by computations of the corresponding quivers and relations for $n\leq 6$, whose apparent regularity suggests an underlying tower, although the algebras are not yet understood sufficiently well to construct them for arbitrary $n$.