Positive weight quadrature on the sphere and monotonicities of Jacobi polynomials

In 2000, Reimer proved that a positive weight quadrature rule on the unit sphere ^d ⊂ ℝ^{d + 1} has the property of quadrature regularity. Hesse and Sloan used a related property, called Property (R) in their work on estimates of quadrature error on ^d. The constants related to Property (R) for a sequence of positive weight quadrature rules on ^d can be estimated by using a variation on Reimer's bounds on the sum of the quadrature weight within a spherical cap, with Jacobi polynomials of the form p^1+d/2,d/2)_tin combination with the Sturm comparison theorem. A recent conjecture on monotonicities of Jacobi polynomials would, if true, provide improved estimates for these constants.