The positivity conjecture for Jacobi polynomial integrals

Let $\alpha,\beta\in(0,\infty)\setminus\mathbb{N}$, and let $F_n^{(\alpha,\beta),\delta}(t)$ denote the Jacobi-polynomial integral defined earlier in the paper. For $t\in(0,\pi]$ and $n\in\mathbb{N}_0$, positivity conjecture.

$$F_n^{(\alpha,\beta),\delta}(t)>0$$

if $\delta\geq\alpha+1$.

This is presented as a more general conjecture than the preceding Gegenbauer case. The paper proves related positivity results under additional parameter conditions and a weaker ceiling threshold, but the assertion for all nonintegral positive $\alpha$ and $\beta$ at the threshold $\delta\geq\alpha+1$ remains open.