Feasibility of probability measures with atoms

Determine which Borel probability measures on [0,∞) with atoms are feasible as distance distributions of complete separable metric probability spaces, beyond the known necessary support condition and its known failure to be sufficient in general.

Background

The paper completely characterizes feasible atomless laws: an atomless probability measure on [0,∞) is feasible exactly when its support contains the origin. The corresponding characterization for measures with atoms remains unresolved. The authors note that the support condition, although necessary for every distance distribution, is not sufficient for arbitrary laws with atoms, as established by a cited counterexample.