Convergence of Conditional Probability a.s.

I am looking for similar results for point-wise convergence of quantile function, but with a different setting.

Suppose $\{B_n\}$ is a series of random variables, $B_n>0$ for each $n$. Let $Z$ be a standard normal variable, and $Z$ is independent from the whole $\{B_n\}$. We know that $B_n\to B$ a.s., where $B$ is a positive definite constant.

Will it be true that$$P\{ B_n Z\leq x \vert B_n\} \to P\{ B Z\leq x\vert B\}$$in some sense (a.s., in probability or other ways), and the limit $P\{ B Z\leq x\vert B\}=P\{ B Z\leq x\}$ since $B$ is a constant matrix?

I appreciate any guidance!