Uniqueness of the stationary distribution for attractive spin systems in a unique stationary environment

Let the background process have a unique stationary distribution. For each environment state $i\in\{1,2\}$, let $c_i(x,\eta)$ be the flip rate at site $x\in\mathbb{Z}$ for configuration $\eta\in\{0,1\}^{\mathbb{Z}}$. Uniqueness conjecture. If

$$c_i(x,\eta)>0$$

for all $x$, $\eta$, and $i=1,2$, then the two extremal stationary distributions coincide:

$$\nu_0=\nu_1.$$

This is proposed as an analogue of Gray's theorem for spin systems without a background process, which gives uniqueness under strict positivity of the rates. The statement concerns whether the same positivity condition ensures uniqueness when the environment evolves randomly.