The algebraic Green-Griffiths-Lang conjecture for complements of very general pairs of divisors

We prove that the complement of a very general pair of hypersurfaces of total degree $2n$ in ${\mathbb{P}}^n$ is algebraically hyperbolic modulo a proper closed subvariety. This provides evidence towards conjectures of Lang-Vojta and Green-Griffiths, and partially extends previous work of Chen, Pacienza-Rousseau, and Chen-Riedl and the third author.