This question is a more specific version of Does there always exist an irreducible representation occurring with multiplicity one when inducing from a closed subgroup to a compact Lie group? . Since that question has already been answered, and I think it would be unfair not to accept the answer given there, I decided to ask the new version in a separate question. (If that is considered bad style, please tell me what to do instead in such a case).

Let $KAN$ be the Iwasawa decomposition of a connected real semisimple Lie group with finite center, and let $M=Z_K(A)$, the centralizer of $A$ in $K$. Let $\tau$ be an irreducible (hence finite-dimensional) complex $M$-representation. Does there exist an irreducible $K$-representation $\sigma$ such that $\tau$ occurs with multiplicity one in $\sigma|_M$?

By Frobenius reciprocity, this is equivalent to the question:

Does the $K$-representation $\mathrm{Ind}^K_M(\tau)$ contain an irreducible subrepresentation that occurs in it only once?

Note that I am not asking for $(K,M)$ to be a Gelfand pair, meaning that the trivial representation occurs with multiplicity one in $\mathrm{Ind}^K_M(\tau)$, or a strong Gelfand pair, meaning that every irreducible representation occurs with multiplicity one in $\mathrm{Ind}^K_M(\tau)$.