Conjecture on stability and convergence of equilibria in the microbial virus model

Assume that $r\kappa\mu_1\neq v\sigma$ and that the $(\operatorname{Coex}_{2,3})$-condition holds, which in particular implies $mv>r+v$. Let $(n_{1a}^*,n_{1i}^*,n_3^*)$ and $(\widetilde n_{2a},\widetilde n_{2d},\widetilde n_{2i},\widetilde n_3)$ denote equilibria of the corresponding three- and four-dimensional virus systems, and let $\mathbf{x}$ denote the coexistence equilibrium of the six-dimensional system. Stability of the equilibria conjecture. The four assertions (A)--(D) in the source hold: asymptotically stable single-host equilibria attract every solution with strictly positive initial conditions; when both are asymptotically stable, every strictly positive solution of the six-dimensional system converges to the equilibrium selected by the invasion conditions, and that equilibrium is unique and asymptotically stable; under the additional condition $\bar n_{2a}<\widetilde n_{2a}$, the corresponding alternatives and uniqueness assertion also hold. These conjectured global convergence and stability statements extend the proved local and invasion results; the source notes that the required equilibrium stability can fail for large $m$ through Hopf bifurcations.