I have a question of a conditioned diffusion processes. This question is somewhat related to an argument which appears in this paper: P.

Let $ D=\{z=(x,y) \in \mathbb{R}^2 \mid |y|<1\}$ and $ K=\{(x,y) \in D \mid x<1\}$ . We denote $ X=(X_t,P_z)$ by the absorbing Brownian motion on $ D$ conditioned to hit $ K$ . We let $ T_{K}=\inf\{t \ge 0 \mid X_t \in K\}$ .

**My question**

We set $ S=\inf\{t \ge 0 \mid \text{ the second coordinate of }X_{t}=-1/2\}$ .

Can we prove the following?: \begin{align*} (1) \quad \lim_{x \to +\infty}\inf_{z=(x,y) \in D \ \text{ with }-1/2<y<1}P_{z}(T_{K}<S)=0. \end{align*}

A claim similar to (1) should hold for more general conditioned diffusions on $ D$ . Can we prove (1) with some universal argument? I would like to know whether similar claims to (1) can be proved for more general conditioned diffusions.