A satisfactory formulation of the laws of physics entails that the future evolution of a physical system should be determined from appropriate initial conditions. The existence of Cauchy horizons in solutions of the Einstein field equations is therefore problematic and expected to be an unstable artifact of general relativity. This is asserted by the strong cosmic censorship conjecture, which was recently put into question by an analysis of the linearized equations in the exterior of charged black holes in an expanding universe. Here, we numerically evolve the nonlinear Einstein-Maxwell-scalar field equations with a positive cosmological constant, under spherical symmetry, and provide strong evidence that mass inflation indeed does not occur in the near extremal regime. This shows that nonlinear effects might not suffice to save the strong cosmic censorship conjecture.