Koolen's conjecture on Terwilliger algebras of finite connected simple graphs

Let $G$ be a finite connected simple graph, and fix a vertex of $G$. The Terwilliger algebra of $G$ is the matrix algebra generated by the adjacency matrix of $G$ and the idempotents corresponding to the distance partition with respect to the fixed vertex.

Koolen's conjecture. For almost all finite connected simple graphs, the Terwilliger algebras coincide with the full matrix algebras.

This conjecture concerns when the distance-based algebra associated with a graph is as large as possible. The source gives no resolution or further qualification of the phrase “almost all,” so the conjecture's precise asymptotic interpretation and current status should be checked.