Beatson–Castell–Xu conjecture on truncated powers on spheres

Let $d\geq 1$, set $\lambda=\frac{d-1}{2}$, and let $({\theta}-t)_+=\max\{{\theta}-t,0\}$. For $\delta\geq\lambda+1$ and any ${\theta}\in(0,\pi)$, define

$$f_{{\theta},\delta}(t)=({\theta}-t)_+^\delta.$$

Beatson–Castell–Xu conjecture. The function $f_{{\theta},\delta}$ is isotropic positive definite on $\mathbb{S}^d$. This conjecture is the positivity assertion underlying the spherical Pólya criterion; the surrounding text indicates that the criterion's proof relies on it, while no resolution is supplied here.