Definition: A $C^*$-algebra $A$ is called sub-homogeneous if there exists $n\in\mathbb{N}$ such that every irreducible representation of $A$ has dimension at most $n$.

I could not find a proof or a counter example for the following claim:

Let $A,B$ be sub-homogeneous $C^*$-algebras and let $C$ be a $C^*$-algebra s.t. there is an exact sequence: $0\to A\to C\to B\to 0$. Does it follow that $C$ is sub-homogeneous?

Thanks