Since the 1970s, it has been known that any open connected manifold of dimension 2, 4 or 6 admits a complex analytic structure whenever its tangent bundle admits a complex linear structure. For half a century, this has been conjectured to hold true for manifolds of any dimension. In this paper, we extend the result to manifolds of dimension 8 and 10. The result is proved by applying Gromov's h-principle in order to adapt a method of Haefliger, originally used to study foliations, to the holomorphic setting. For dimension 12 and greater, the conjecture remains open.