For a measurable space $(E,\mathcal E)$ and a Markov kernel $P:E\times \mathcal E\to[0,1]$ there is a unique homogeneous Markov chain $X$. The first return time is defined as $ \tau_A = \inf\{k\geq 1:X_k\in A\}. $ Denote $L(x,A) = \mathbb P_x[\tau_A<\infty]$ for all $x\in E$ and $A\in \mathcal E$.
I have the following question: is there $\delta(P)>0$ such that for all Markov kernels $Q$ satisfying $ \|P-Q\| = \sup\limits_{\begin{align}x&\in E\\B&\in \mathcal E\end{align}}|P(x,B)-Q(x,B)|<\delta $ the property $L_P(x,A) =1$ for all $x\in E$ and some $A$ implies $L_Q(x,A) = 1$ for all $x\in E$?