Conjecture on stochastic invasion of type 2

Let $(n_{1a}^*,n_{1i}^*,n_3^*)$ be an asymptotically stable equilibrium of the three-dimensional virus system, and let $(\widetilde n_{2a},\widetilde n_{2d},\widetilde n_{2i},\widetilde n_3)$ be an asymptotically stable equilibrium of the four-dimensional virus system. For an equilibrium $\widehat{\mathbf n}$ and $\delta>0$, let $T_{\widehat{\mathbf n},\delta}$ be the first time the individual-based process enters its $\delta$-neighbourhood, and let $T^{\mathrm{fix}}$ additionally require extinction of the losing host type. Type 2 invasion conjecture. For all sufficiently small $\delta>0$, if type 2 invades type 1 but type 1 does not invade type 2, then, starting from the type-1 equilibrium, fixation at the type-2 equilibrium before type-2 extinction or coexistence has limiting probability $1-s_{2a}$, and conditional on reaching the type-2 neighbourhood, the fixation time divided by $\log K$ converges in probability to $1/\widetilde\lambda+1/\lambda^*$. If both invasion directions are possible, reaching the coexistence neighbourhood before type-2 extinction or fixation at the type-2 equilibrium has limiting probability $1-s_{2a}$, and the corresponding time divided by $\log K$ converges in probability to $1/\widetilde\lambda$. If additionally $tilde n_{2a}<\bar n_{2a}$ and type 2 invades type 1, the displayed conclusions concerning $s_{2a}$ and $tilde\lambda$ also hold. These statements are stochastic counterparts of the deterministic invasion picture; their validity remains conjectural.