Differentially positive systems are the nonlinear systems whose linearization along trajectories preserves a cone field on a smooth Riemannian manifold. One of the embryonic forms for cone fields in reality is originated from the general relativity. Out of a set of measure zero, we show that almost every orbits will converge to equilibrium. This solved a reduced version (from a measure-theoretic perspective) of Forni-Sepulchre's conjecture in 2016 for globally orderable manifolds.