In this question, I'm borrowing the notations from Minguez' paper on unramified representations of unitary groups. Let $F$ be a $p$-adic field and let $G$ be a connected reductive group over $F$. Let $P_0$ be a minimal $F$-parabolic subgroup and $M_0$ a Levi factor of $P_0$ defined over $F$. Let $W := \mathrm{N}_G(M_0)/M_0$ denote the spherical Weyl group of $G$. We denote by $X^{\mathrm{un}}(M_0)$ the set of unramified characters of $M_0$, that is those (complex) characters of $M_0$ which are trivial on the unique maximal compact subgroup of $M_0$. The Weyl group $W$ acts on $X^{\mathrm{un}}(M_0)$ through the formula $(w\chi)(m) := \chi(w^{-1}mw)$.

We give ourselves a good special maximal compact subgroup $K$ in $G$. A smooth representation of $G$ is called $K$-spherical if it contains a non-zero $K$-invariant vector. Then the set of isomorphism classes of $K$-spherical representations is in bijection with the quotient set $X^{\mathrm{un}}(M_0)/W$.

Such a bijection is induced by the following map $X^{\mathrm{un}}(M_0) \rightarrow \{K\text{-spherical representations}\}$. Let $\chi \in X^{\mathrm{un}}(M_0)$ and denote by $\iota_{P_0}^G\,\chi$ the normalized parabolic induction of $\chi$. The space of $K$-invariant vectors in $\iota_{P_0}^G\,\chi$ has dimension $1$ and therefore, there is a unique composition series $\pi_{K,\chi}$ of $\iota_{P_0}^G\,\chi$ which is $K$-spherical. The map is then $\chi \mapsto \pi_{K,\chi}$.

My question is about what happens if I allow $K$ to vary among all good special maximal compact subgroups of $G$. For instance, if $G$ is a unitary group then there are two distinct conjugacy classes of such subgroups.

If $K'$ is another such subgroup, are $K$-spherical and $K'$-spherical representations the same thing ? In other words, are the representations $\pi_{K,\chi}$ and $\pi_{K',\chi}$ the same ? Or is there a concrete counterexample ?

Unless I'm mistaken, if $\pi$ is a character of $G$ then it is $K$-spherical if and only if it is $K'$-spherical ; at least it seems to be the case for unitary groups. But what about higher dimensional representations ?