Mixed-equilibrium conjecture for multiple pathogens

Consider the multi-species SLIR system for $V$ pathogens and $P$ species, with pathogens indexed by $a=1,\dots,k$ satisfying $\mathcal{R}_{0a}<1$ and pathogens indexed by $e=k+1,\ldots,V$ satisfying $\mathcal{R}_{0e}>1$, for some $k\in\{1,\ldots,n\}$. Mixed-equilibrium conjecture. The system has an unstable mixed equilibrium in which pathogens $1,\ldots,k$ are absent and pathogens $k+1,\ldots,V$ are present at an endemic level. This conjecture is presented as a natural extension of the paper's theorem on the existence of mixed endemic equilibria and concerns the apparent partial independence of pathogens; its resolution is not established in the supplied text.