Continuation : Does the density of a stopped drifted Brownian motion vanish at zero?

Let

$$Y_t:=1+\int_0^t b_sds + W_t,\quad\forall t\ge 0,$$

where $$(b_t)_{t\ge 0}$$ is a bounded adapted process and $$(W_t)_{t\ge 0}$$ is a standard Brownian motion. Denote $$\tau:=\{t\ge 0: Y_t\le 0\}$$ and $$X_t:=Y_{t\wedge \tau}$$. It is known that the law of $$X_t$$, denoted by $$\mu_t$$, has the following decomposition:

$$\mu_t(dx) = \alpha(t)\delta_0(dx) + p_t(x)dx,\quad \forall t>0.$$

In the previous post Does the density of a stopped drifted Brownian motion vanish at zero? Mateusz has shown $$p_t(0+):=\lim_{x\to 0+}p_t(x)=0$$ for $$t>0$$ when the drift $$b$$ is deterministic. Can we generalize this result to adapted processes?

PS : Mateusz claims that using Chapman-Kolmogorov equation and a comparison argument allows to conclude that the density $$X_t$$ goes to $$0$$ at zero, but I don't know the details.