Let $G$ be a simple Lie group of Hermitian type, and $G'$ be a reductive subgroup of $G$. Suppose that $G'$ is also of Hermitian type and contains the center of the maximal compact subgroup of $G$. Thus it is well known that any irreducible (anti)-holomorphic representation (or highest / lowest weight representation) of $G$ is discretely decomposable upon restriction to $G'$.

Now the question is: if $\tau$ is an irreducible (anti)-holomorphic representation of $G'$, does there exist an irreducible (anti)-holomorphic representation $\pi$ of $G$ such that $\tau$ is embedded into $\pi$ as $G'$-representations?

In order words, do the restrictions of irreducible (anti)-holomorphic representations of $G$ exhaust all the irreducible (anti)-holomorphic representations of $G'$?