Let $ μ_1 $ and $ μ_2 $ be two, in general complex-valued, Borel measures on the real line such that $ \mathrm{supp} \,μ_1 =[α_1,β_1] < \mathrm{supp}\,μ_2 =[α_2,β_2] $ and $ dμ_i(x) = -ρ_i(x)dx/2π\mathrm{i} $, where $ ρ_i(x) $ is the restriction to $ [α_i,β_i] $ of a function non-vanishing and holomorphic in some neighborhood of $ [α_i,β_i] $. Strong asymptotics of multiple orthogonal polynomials is considered as their multi-indices $ (n_1,n_2) $ tend to infinity in both coordinates. The main goal of this work is to show that the error terms in the asymptotic formulae are uniform with respect to $ \min\{n_1,n_2\} $.