An invariant Gibbs target can still give reducible dynamics

Each coordinate marginal under $\pi$ is uniform on $\{0,1\}$. Given $Y=y$, the conditional law of $X$ is the point mass ${\delta }_y$; given $X=x$, the conditional law of $Y$ is ${\delta }_x$. These are valid full conditionals because the target assigns mass only to the two diagonal points.

From $(0,0)$ either coordinate update leaves the state at $(0,0)$; from $(1,1)$ either update leaves it at $(1,1)$. The random-scan mixture in [F3] therefore makes each singleton a closed class. The target is invariant by [F4], but $\{(0,0)\}$ is nonempty and proper, so reducibility follows from [F5].

Starting from $(0,0)$, the time-$n$ law is ${\delta }_{(0,0)}$, whose total-variation distance from $\pi$ is $1\mathrm{/}2$. Thus target invariance alone does not imply convergence from every start, uniqueness of the invariant law, or irreducibility.