Watkins' GRR existence conjecture for sufficiently large groups

Let $G$ be a group, and call a group exceptional if it does not admit a graphical regular representation (GRR), excluding the two known obstructions: being abelian of exponent greater than $2$ or being generalized dicyclic.

Watkins' conjecture. There is an integer $n$ such that every group of cardinality at least $n$ which is neither abelian of exponent greater than $2$ nor generalized dicyclic admits a GRR.

This is the quantitative formulation of Watkins' assertion that, outside the two infinite exceptional families, only finitely many groups fail to admit GRRs. The supplied source does not state that it has been resolved.