Other natural measures with equal three- and five-vertex probabilities

Determine whether any other natural probability measures, besides the planar measure with density 1/[π(1+x²+y²)²], yield equal probabilities that the convex hull of five independently sampled points has three and five vertices, respectively.

Background

The paper studies five points sampled independently from the planar probability measure with density 1/[π(1+x²+y²)²], obtained by stereographically projecting uniformly distributed points from the unit sphere. For this measure, the probabilities p₃ and p₅ that the convex hull has three and five vertices are equal. The authors ask whether this equality occurs for any other natural probability measures.

The discussion identifies beta-prime distributions and beta distributions as natural families of measures that could be considered in connection with this question, although the paper does not resolve whether the equality p₃ = p₅ holds for any additional measures.