Determine whether rainbow and list rainbow connection numbers always coincide in the diameter-two case

Determine whether every connected graph $G$ of diameter two satisfies $rc(G)=rc^\ell(G)$, including the unresolved possibility that $rc(G)=2$ and $rc^\ell(G)=3$ for graphs with a universal vertex.

Background

The paper investigates the broader equality $rc(G)=rc^\ell(G)$ and analyzes diameter-two graphs as a potentially tractable case. For graphs with a universal vertex, the authors prove that equality holds except for the possible values $rc(G)=2$ and $rc^\ell(G)=3$. For diameter-two graphs without a universal vertex, they discuss sparse extremal graphs and show equality for a particular minimum-edge family, but do not resolve the general case.