I will be very brief.

Let $A, C$ be two bounded operators on a Hilbert space $\mathcal{H}$, such that both $AC$ and $CA$ are trace-class operators. Let $B_{n}$ be a sequence of bounded operators such that:

$i)$ $B_{n}\to I$ as $n\to \infty$ strongly, where $I$ is the identity operator;

$ii)$ $AB_{n}C$ is a trace-class operator for any $n \in \mathbb{N}$.

Could someone tell me if it holds that $AB_{n}C\to AC$ in trace-class sense? If we replace strong convergence, by norm convergence, does the statement hold?