In Theorem 4.9 of the book Markov Chains and Mixing Times by Levin & Peres, there is a convergence theorem for ergodic Markov chains which states that there exist constants $C > 0$ and $\alpha \in (0,1)$ such that $$\sup_{x \in \mathcal{X}}\|P^t(x) - \pi\|_\text{TV} \leq C \alpha^t$$
Here $\mathcal{X}$ is the state space of the Markov chain, $P$ is a time independent transition kernel, and $\pi$ is its stationary distribution. A Markov chain that has this convergence rate is called geometric uniform ergodic.
Is this the fastest rate at which a Markov chain can converge? If true, is there a way to prove it? If false, are there examples of chains that converge at a rate faster than geometric?
Edit: I realize that i.i.d. chains have "already" converged, so they're obviously faster than an ergodic MC. However, I'm looking for examples with a non-trivial transition kernel, ideally a class of Markov chains that converge faster than a geometric rate.