Erdős Problem 397

References:

• erdosproblems.com/397

theorem Erdos397.erdos_397 : {(M, N) : Finset ℕ × Finset ℕ | Disjoint M N ∧ ∏ i ∈ M, i.centralBinom = ∏ j ∈ N, j.centralBinom}.Finite ↔ sorry

Are there only finitely many solutions to $$ \prod_i \binom{2m_i}{m_i}=\prod_j \binom{2n_j}{n_j} $$ with the $m_i,n_j$ distinct?