Holomorphic curves omitting five planes in projective space

In 1928 H. Cartan proved an extension of Montel's normality criterion to holomorphic curves in the complex projective plane P². He also conjectured that a similar result is true for holomorphic curves in Pⁿ for any n. Recently, the author constructed a counterexample to this conjecture for any n ≥ 3. In this paper we show how to modify Cartan's conjecture so that it becomes true, at least for n = 3.