The positivity conjecture for integrals of Gegenbauer polynomials

Let $\delta>0$, $\lambda>0$, and $n\in\mathbb{N}_0$. For every $0<t\leq\pi$, define

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

Positivity conjecture. $F_n^{\lambda,\delta}(t)>0$ for every $t\in(0,\pi]$ if $\delta\geq\lambda+1$.

This conjecture would yield strict positive definiteness of the truncated power functions $(t-\theta)_+^\delta$ on spheres in the range needed for the Polyá-type criterion, including odd dimensions. The paper proves the weaker threshold $\delta\geq\lceil\lambda\rceil+1$, while the improvement to $\delta\geq\lambda+1$ remains open in the stated context.