List rainbow connection number of graphs with rainbow connection number two

Determine whether every connected graph G satisfying rc(G)=2 also satisfies rc^ℓ(G)=2.

Background

The paper establishes that rc(G)=2 if and only if src(G)=2, and that src^ℓ(G)=2 implies rc^ℓ(G)=2, which in turn implies rc(G)=2 and src(G)=2. The converse implication from rc(G)=2 to rc^ℓ(G)=2 is not established.

The authors indicate that this issue will be discussed further in the section concerning prescribed values of the rainbow connection parameters, where they investigate possible separations between ordinary and list versions.