Conjecture on extinction and non-halting in multitype branching processes in a Markovian environment

Let $G=(V,E)$ be a finite directed graph. For each vertex $v\in V$, let $S_v$ be a finite state space, let $P_v$ be an irreducible stochastic transition matrix, and let $R_v$ be a stochastic reproduction matrix describing the offspring sent to the out-neighbors of $v$. Let $\pi_v$ be the stationary distribution of $P_v$, let $\mu_{vw}(i)$ be the mean number of offspring sent from $v$ to $w$ when the state of $v$ is $i$, and define

$$M_{vw}:=\sum_{i\in S_v}\pi_v(i)\mu_{vw}(i).$$

Let $\mu$ be the Perron–Frobenius eigenvalue of the matrix $M$. Multitype BPME halting conjecture. If $\mu<1$, then the network halts almost surely for any initial state and population. If $\mu=1$ and there are no conserved quantities, then the network halts almost surely for any initial state and population. If $\mu>1$, then for sufficiently large initial population the network has a positive probability not to halt. The claim proposes a threshold criterion for almost-sure halting versus positive-probability non-halting in a network of branching processes in a Markovian environment. No resolution is supplied in the source.