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