If $X$ is a right-continuous process, is $t\mapsto\operatorname E\left[X_\tau\mid\tau=t\right]$ right-continuous as well?

Question

Let

• $(\Omega,\mathcal A,\operatorname P)$ be a probability space;
• $(X_t)_{t\in[0,\:\infty]}$ be a real-valued process on $(\Omega,\mathcal A,\operatorname P)$;
• $\tau$ be an $[0,\infty]$-valued random variable on $(\Omega,\mathcal A,\operatorname P)$.

Under which assumptions are we able to show that if $(X_t)_{t\in[0,\:\infty]}$ is right-continuous at $0$, then $$\varphi(t):=\operatorname E\left[X_\tau\mid\tau=t\right]\;\;\;\text{for }t\in[0,\infty]$$ is right-continuous at $0$ as well?

We clearly need to assume that $X:\Omega\times[0,\infty]\to\mathbb R$ is $(\mathcal A\otimes\mathcal B([0,\infty]),\mathcal B(\mathbb R))$-measurable in order to know that $X_\tau$ is $\mathcal A$-measurable. Moreover, we clearly need to assume that $X_t\in\mathcal L^1(\operatorname P)$ so that $\varphi$ is actually well-defined.