Regularity conjecture for uniquely $K_r$-saturated graphs

Let $r$ be a positive integer. A graph is uniquely $K_r$-saturated if it is $K_r$-free and adding any missing edge creates exactly one copy of $K_r$; a dominating vertex is a vertex adjacent to every other vertex. A graph is regular if all its vertices have the same degree.

Regularity conjecture. For each $r$, a uniquely $K_r$-saturated graph with no dominating vertex is regular.

All examples known in the source are regular, which motivates this conjecture. Its resolution is not supplied and remains open in the paper's context.