Consider the 1-dimensional Ising model on a torus (or a line of which the endpoints are connected) of size N. Suppose that the configuration evolves according to the heat-bath dynamics, that is, each time step one of the N spins is selected at random and it takes the value +1 with probability $p = \dfrac{1}{1+\exp{(-2T\sum\limits_{j \in \delta_i}\sigma(j)})}$ and the value -1 with probability $1-p$, where $\sigma$ is the current configuration, $i$ is the selected spin and $\delta_i$ is the neighbourhood of spin $i$.
Suppose that the all-plus configuration, denoted by $\sigma^+$ is an absorbing configuration. Furthermore, we say that a configuration $\sigma$ is dominated by a configuration $\eta$, denoted by $\eta > \sigma$ if $\sigma(i) = +1$ implies $\eta(i) = +1$ for each spin $i$ and if there is at least one spin $j$ such that $\eta(j) = +1$ and $\sigma(j) = -1$. Now consider 5 configurations $\sigma^1, \sigma^2, \sigma^3, \sigma^4$ and $\sigma^5$ that satisfy $\sigma^1 > \sigma^2 > \sigma^3, \sigma^4, \sigma^5$. I am now trying to prove that $P(\sigma_t = \sigma^+| \sigma_0 = \sigma^1) + P(\sigma_t = \sigma^+ | \sigma_0 = \sigma^3) + P(\sigma_t = \sigma^+| \sigma_0 = \sigma^4) \geq P(\sigma_t = \sigma^+| \sigma_0 = \sigma^2) + P(\sigma_t = \sigma^+ | \sigma_0 = \sigma^2) + P(\sigma_t = \sigma^+| \sigma_0 = \sigma^5)$ for each time step $t = 1, 2, ...$.
I think a proof involving some clever coupling and a stochastic domination argument should do the job here, but I am not sure how to proceed. Help would be greatly appreciated!