Possible pairs of ordinary and list rainbow connection numbers

Characterise all pairs of positive integers a and b for which there exists a connected graph G with rc(G)=a and rc^ℓ(G)=b, and determine whether rc(G)=rc^ℓ(G) for every connected graph G.

Background

The inequalities rc(G)≤rc^ℓ(G) and the characterization of graphs with parameter value one show that any realizable pair must satisfy a=b=1 or 2≤a≤b. The paper proves analogous complete realizability results for pairs involving src(G), src^ℓ(G), and for rc^ℓ(G), src^ℓ(G), but does not obtain such a result for rc(G), rc^ℓ(G).

In particular, the authors report that they have not constructed a connected graph with rc(G)=a< b=rc^ℓ(G) for any 2≤a<b. The broader equality question is presented as analogous to the list colouring conjecture.