A Markov Chain on a measurable space $X$ is uniquely determined by a stochastic kernel $P$ on $X$. Let $\mathsf P_x$ denote the probability on paths generated by $P$ and the initial condition $x\in X$. It is easy to come up with examples of $P$ such that for any $\delta>0$ there exists $\bar P$ satisfying $$ \|P - \bar P\| := \sup_{x\in X}\|P_x - \bar P_x\| \leq \delta $$ but $\|\mathsf P_x - \bar{\mathsf P}_x\| = 1$ for some $x\in X$, where $\|\cdot\|$ denotes the total variation metric, that is measures $\mathsf P_x$ and $\bar {\mathsf P}_x$ are mutually singular.
I wonder, whether the above statement holds for every Markov Chain, or for some $P$ and $\delta >0$, for all $\bar P$ satisfying $\|P - \bar P\|\leq \delta$ it holds that $\sup_{x\in X}\|\mathsf P_x - \bar{\mathsf P}_x\| <1$. Clearly, I assume that $X$ has at least two elements to avoid a trivial example of a singleton.
In most cases, $\mathsf P_x$ and $\bar {\mathsf P}_x$ are mutually singular. For example, in the finite irreducible case, fixing two states $x$ and $y$ and some initial distribution $\nu$, the ergodic theorem shows that $N^n_{xy}/N^n_x\to P(x,y)$ almost surely for $\mathsf P_\nu$ when $n\to\infty$ and that $N_{xy}/N_x\to \bar P(x,y)$ almost surely for $\bar {\mathsf P}_\nu$ when $n\to\infty$, where $N^n_x$ denotes the number of visits of $x$ before time $n$ and $N^n_{xy}$ the number of steps from $x$ to $y$ before time $n$. As soon as $P(x,y)\ne\bar P(x,y)$ for at least one pair $(x,y)$, that is, as soon as $P\ne\bar P$, this proves that $\mathsf P_\nu$ and $\bar {\mathsf P}_\nu$ are mutually singular.