Bisecting d+2 mass distributions with a fixed similar copy

Determine whether there exists a compact set, possibly star-shaped, in R^d such that every collection of d+2 mass distributions on R^d can be simultaneously bisected by a similar copy of that set.

Background

The paper proves that d+1 mass distributions in R^d can be simultaneously bisected by suitable similar copies of a broad family of compact cookie cutters. The conclusion asks whether the number d+1 can be increased to d+2 for at least one compact set, potentially one that is star-shaped.

For convex sets, the authors explain that d+2 simultaneous bisections are impossible in general, using essentially point-like masses with one mass inside the convex hull of the others. The unresolved issue is therefore whether some nonconvex or specially structured compact set can achieve this stronger universal bisection property.