Asymptotic shape conjecture for the contact process in an evolving random environment

Let $(\mathbf{C},\mathbf{B})$ be a contact process in an evolving random environment (CPERE) on the $d$-dimensional integer lattice, with infection rate $\lambda>0$ and recovery rate $r>0$. Assume that $\mathbf{B}$ satisfies Assumption~, that $\theta(\lambda,r,\{\mathbf{0}\},\emptyset)>0$, and that there are constants $C_1,C_2,M>0$ satisfying

$$\mathds{P}(t\leq \tau<\infty)\leq C_1\exp(-C_2t),$$ $$\mathds{P}(x\notin \mathbf{H}_{M\lVert x\rVert_1+t},\tau=\infty)\leq C_1\exp(-C_2t),$$ $$\mathds{P}(x\notin \mathbf{K}_{M\lVert x\rVert_1+t},\tau=\infty)\leq C_1\exp(-C_2t).$$

Here $\tau$ is the extinction time, $\mathbf{H}_t$ is the set of vertices infected by time $t$, and $\mathbf{K}_t$ is the permanently coupled region; write $\mathbf{H}'_t=\mathbf{H}_t+[-\tfrac12,\tfrac12]^d$ and $\mathbf{K}'_t=\mathbf{K}_t+[-\tfrac12,\tfrac12]^d$.

Asymptotic shape conjecture. There exists a bounded convex set $U\subset\mathds{R}^d$ such that, for every $\varepsilon>0$,

$$\mathds{P}\left(\exists s\geq0:\ t(1-\varepsilon)U\subset(\mathbf{K}'_t\cap\mathbf{H}'_t)\subset\mathbf{H}'_t\subset t(1+\varepsilon)U\ \forall t\geq s\mathrel{\Big|}\tau=\infty\right)=1.$$

This is the expected asymptotic shape theorem for the CPERE: conditional on survival, the infected region and its permanently coupled core should grow with a deterministic convex shape. Analogous results are known for contact processes in static random environments and for contact processes with ageing, while this formulation remains conjectural here.