Positive-definite function classification conjecture for finite spheres

Let $n>2$, let $f:[-1,1]\to\mathbb{R}$ satisfy $f(1)=1$, and define $\widehat f:SO(n)\to\mathbb{R}$ by $\widehat f(\gamma)=f(\gamma_{11})$. Assume that $\widehat f$ defines a positive definite function on the homogeneous space

$$SO(n)/SO(n-1)\cong S^{n-1}.$$

Here $\chi$ is the indicator function and $P^{(n-1)}$ denotes the relevant Gegenbauer polynomial. Positive-definite function classification conjecture. The function $f$ is of the form

$$f(x)=a\chi(x=1\vee x=-1)+bx\chi(x=1\vee x=-1)+\sum_i c_iP^{(n-1)}(x),$$

where $a,b,c_i\geq 0$ and $a+b+\sum_i c_i=1$. This is presented as a conjectural corollary of the claimed classification of class-one representations and would classify positive definite functions on the sphere arising from $SO(n)/SO(n-1)$. The source gives no resolution; the surrounding discussion connects the question to possible pathological positive definite functions and finite constrained-angle codes.