Damase's continuity approximation conjecture for positive definite functions on spheres

Let $S^{n-1}$ be the unit sphere, and let a positive definite function on $S^{n-1}$ be understood in the usual sense for functions on the sphere. Damase's continuity approximation conjecture. Any positive definite function on $S^{n-1}$ is a pointwise limit of continuous ones. The conjecture concerns whether continuity can be recovered by pointwise approximation within the class of positive definite functions; the supplied text does not state a resolution.