Reduction from bounded-color thin set principles

Determine whether, for every pair of natural numbers n, ell >= 1, there exists a natural number m such that RCA_0 proves RT_{ell+1,ell}^{n+1} -> RT_{<infty,m}^n.

Background

The paper asks whether the principle RT_{ell+1,ell}^{n+1}, asserting that every coloring of (n+1)-tuples with ell+1 colors has an infinite set using at most ell colors, uniformly implies a thin set theorem RT_{<infty,m}^n for some finite m depending on n and ell.

An affirmative answer would, together with the paper’s main theorem, yield RT_{ell+1,ell}^{n+1} -> B Sigma_{n+1}^0 over RCA_0. The question is nontrivial because although it is automatic when RT_{ell+1,ell}^{n+1} implies ACA_0, there are known instances where the former is weaker than ACA_0 and the desired reduction nevertheless holds.