A Pólya integral positivity conjecture for Gegenbauer polynomials

Let $b4>0$, $bb>0$, and $n\in\mathbb{N}_0$. For $0<t<\pi$, define

$$F_n^{\lambda,\delta}(t)=\int_0^t (t-\theta)^\delta C_n^\lambda(\cos\theta)(\sin\theta)^{2\lambda}\,d\theta.$$

Integral positivity conjecture. The inequality $F_n^{\lambda,\delta}(t)>0$ holds for every $n\in\mathbb{N}_0$ and every $t\in(0,\pi]$ if and only if $\delta\geq\lambda+1$.

This non-negativity statement is the key ingredient in extending the stated Pólya-type criterion for positive definite and strictly positive definite zonal functions from dimensions $3\leq d\leq 8$ to all dimensions $d>2$.