Let $(f_{\epsilon})_{\epsilon>0}$ be a family of non-increasing and continuous functions on $\mathbb R_+$ s.t. $f_{\epsilon}(0)=1$ and $f_{\epsilon}(\infty)=0$. Assume that $\epsilon\mapsto f_\epsilon(t)$ is non-increasing for all $t\ge 0$ and denote by $f:\mathbb R_+ \to [0,1]$ the pointwise limit of $(f_{\epsilon})_{\epsilon>0}$. For every $t\ge 0$, set

$$X^{\epsilon}_t=X_0+W_t-1+f_\epsilon(t) \quad\mbox{and} \quad X_t=X_0+W_t-1+f(t),$$

where $X_0>0$ is a random variable and $(W_{t})_{t\ge 0}$ is an independent Brownian motion. Can we prove

$$\lim_{\epsilon\to 0+} \left\{ \mathbb E \left[ \exp\left(-\frac 1 \epsilon \int_0^t \big(X_s^{\epsilon}\big)^-ds\right)\right] - \mathbb E \left[ \exp\left(-\frac 1 \epsilon \int_0^t \big(X_s\big)^-ds\right)\right]\right\} = 0$$

holds for almost every $t \ge 0$? Here $a^-:=\max(-a, 0)$ for $a\in\mathbb R$.