# multiplicity one for restriction of representations from $GU$ to unitary group

Let $E/F$ be a quadratic field extension of p-adic fields. Let $V$ be a (skew-)Hermitian space and $U(V)$ be the unitary group. Let $GU(V)$ be the similitude unitary group. Given an irreducible smooth representation $\pi$ of $GU(V)$, do we know that the restriction $\pi|_{U(V)}$ has multiplicity one?

For the pair $(GL(n),SL(n))$ similar results are proved by Tadic. For the pair $(GSp, Sp)$, similar results are proved by Adler-Prasad. I am wondering if the unitary group version is true or not.

Thanks.

In case you didn't already see it, this question has now been answered in the affirmative by Adler and Prasad in Theorem 12a here: $$[$$1$$]$$
$$[$$1$$]$$ Jeffrey D. Adler, Dipendra Prasad. Multiplicity upon restriction to the derived subgroup, 2018. (arXiv link)