Naive classification conjecture for class-one representations of $SO(n)$

Let $SO(n)$ be the special orthogonal group, and let $SO(n-1)$ be the subgroup fixing the first coordinate. A class-one representation means an irreducible representation admitting a nonzero $SO(n-1)$-invariant vector. For an element $\gamma\in SO(n)$, write $\gamma_{11}$ for its upper-left entry, let $P^{(n-1)}$ denote the relevant Gegenbauer polynomial, and let $\chi$ denote the indicator function. Naive classification conjecture. Every irreducible class-one representation of $SO(n)$ arises from one of the states

$$\varphi(\gamma)=P^{(n-1)}(\gamma_{11}),\qquad \varphi(\gamma)=\chi(\gamma_{11}=1\vee\gamma_{11}=-1),$$

or

$$\varphi(\gamma)=\gamma_{11}\chi(\gamma_{11}=1\vee\gamma_{11}=-1).$$

This would classify the possible invariant states underlying class-one representations and clarify whether discontinuous representations produce additional positive definite functions. The source presents this explicitly as a naive conjecture and gives no resolution.