Let $G$ be a noncompact real simple Lie group not of Hermitian type, and $\mathfrak{g}_0$ its Lie algebra. Fix a maximal compact subgroup $K$ in $G$ with its Lie algebra $\mathfrak{k}_0$. Write $\mathfrak{g}_0=\mathfrak{k}_0+\mathfrak{p}_0$ for the Cartan decomposition, and $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ for complexification. Fix a Cartan subalgebra $\mathfrak{t}$ in $\mathfrak{k}$ and a positive system for $(\mathfrak{k},\mathfrak{t})$.
If $\pi$ is the minimal representation of $G$, i.e., its annihilator ideal in the universal enveloping algebra $U(\mathfrak{g})$ is the Joseph ideal, then it is well known that the $K$-types of $\pi$ lie in the set in the form of $\{\mu+t\beta\mid t\in\mathbb{Z}_{\geq0}\}$ for some $\mu\in\mathfrak{t}^*$, where $\beta$ is the highest weight of $\mathfrak{p}$ as the $K$-representation.
Now suppose that $\pi$ is a representation of $G$ such that its Gelfand-Kirillov dimension is equal to half of the dimension of the minimal nilpotent orbit under the adjoint action of $G$ on $\mathfrak{g}$, but $\pi$ is not a minimal representation of $G$. Do the $K$-types of $\pi$ still lie in the set in the form of $\{\mu+t\beta\mid t\in\mathbb{Z}_{\geq0}\}$?
