Spiga's cubic GRR conjecture with one involution

Let $q$ be a prime power. A cubic graphical regular representation (cubic GRR) of a finite group is a cubic Cayley graph whose automorphism group is the right regular representation of the group; its connection set is the inverse-closed generating set defining the Cayley graph. An involution is an element of order two.

Spiga's conjecture. Except for a finite number of cases and for the groups $\operatorname{PSL}_2(q)$, every finite nonabelian simple group has a cubic GRR whose connection set contains exactly one involution.

The conjecture strengthens the finiteness conjecture by prescribing the number of involutions in the connection set. The source says that known sufficient conditions support it, but does not establish the full assertion.