Monotonicity of weak rainbow saturation across trees

Determine whether every tree T on ℓ vertices satisfies rwsat(n,P_ℓ) ≤ rwsat(n,T) ≤ rwsat(n,S_ℓ) for some sufficiently large n.

Background

The paper establishes that, for sufficiently large n and ℓ in the stated ranges, the path P_ℓ has weak rainbow saturation number ℓ+1, while the star S_ℓ has weak rainbow saturation number \binom{ℓ}{2}-1. Paths and stars are extremal among trees for ordinary weak saturation numbers, motivating the question of whether the same ordering holds for weak rainbow saturation numbers across all trees of order ℓ.

The unresolved problem asks whether every intermediate tree T is bounded between the path and star values for at least some sufficiently large n. The statement does not assert a uniform threshold in ℓ or an inequality for all sufficiently large n, only the existence of a sufficiently large n for each tree.