Every nonflat conformal minimal surface is homotopic to a proper one

Given an open Riemann surface M, we prove that every nonflat conformal minimal immersion Mⁿ (n 3) is homotopic through nonflat conformal minimal immersions Mⁿ to a proper one. If n 5, it may be chosen in addition injective, hence a proper conformal minimal embedding. Prescribing its flux, as a consequence, every nonflat conformal minimal immersion Mⁿ is homotopic to the real part of a proper holomorphic null embedding Mⁿ. We also obtain a result for a more general family of holomorphic immersions from an open Riemann surface into Cⁿ directed by Oka cones in Cⁿ.