Generical behavior of flows strongly monotone with respect to high-rank cones

We consider a $C^{1,\alpha}$ smooth flow in $\mathbb{R}^n$ which is "strongly monotone" with respect to a cone $C$ of rank $k$, a closed set that contains a linear subspace of dimension $k$ and no linear subspaces of higher dimension. We prove that orbits with initial data from an open and dense subset of the phase space are either pseudo-ordered or convergent to equilibria. This covers the celebrated Hirsch's Generic Convergence Theorem in the case $k=1$, yields a generic Poincar\'{e}-Bendixson Theorem for the case $k=2$, and holds true with arbitrary dimension $k$. Our approach involves the ergodic argument using the $k$-exponential separation and the associated $k$-Lyapunov exponent (that reduces to the first Lyapunov exponent if $k=1$).