Conjecture on the limiting stochastic fixed point of the branching-process generating function

Let $S$ be the environment state space, let $f(M)=\sum P_nM^n$ be the matrix generating function, and let $E$ be the extinction matrix. Write $I$ for the identity matrix, and let $v$ be the left Perron–Frobenius eigenvector of $E$, normalized so that its coordinates sum to $1$. Limiting stochastic fixed-point conjecture. For all environment states $i,j\in S$,

$$\lim_{n \to \infty} f^n(I)_{ij} = E_{ij} + \left(1-\sum_{k \in S} E_{ik}\right) v_j.$$

The conjecture expresses the expected limiting distribution of the environment after $n$ generations on the event of survival. The paper motivates it with experiments; no resolution is supplied.