Finiteness conjecture for $r$-primitive graphs

Let $r\geq 3$. An $r$-primitive graph is a uniquely $K_r$-saturated graph with no dominating vertex. Finiteness conjecture. For each $r\geq 3$, there are a finite number of $r$-primitive graphs. The major open question is whether the number of $r$-primitive graphs is finite for every fixed $r$; known examples include complements of odd cycles and several sporadic graphs, and regularity is not necessary.