This question can be seen as a variant of the post Bounded density for diffusions with diffusion coefficients bounded away from $0$ by Iosif Pinelis. Namely, consider the diffusion

$$X_t=\int_0^t a(s,X_s){\bf 1}_{\{|X_s|<1\}}dW_s,\quad \forall t\ge 0,$$

where $(W_t)_{t\ge 0}$ is a standard Brownian motion and $a$ is smooth s.t. $\inf_{(t,x)}a(t,x)\ge c>0$. Denote by $\mu_t$ the distribution of $X_t$. Can we write (under suitable conditions)

$$\mu_t(dx) = q^+_t\delta_{1}(dx)+q^-_t\delta_{-1}(dx)+ p_t(x)dx?$$

Here $q^{\pm}_t=\mathbb P[X_t=\pm 1]$ and $p_t$ (up to a normalization) is the conditional density function of $X_t$ knowing $\{\tau>t\}$, where $\tau:=\inf\{t\ge 0: |X_t|\ge 1\}$. So an alternative formulation is whether $X_t$ admits a density on the event $\{\tau>t\}$.

Any solution, references or comments are appreciated.