The discontinuity conjecture for free compound Poisson distributions

Let $u$ be a probability measure such that $u(\{0\})=0$. The free infinitely divisible measure ${\bm \pi} \boxtimes \nu$ is absolutely continuous with respect to Lebesgue measure, and its density is discontinuous at $0$; more strongly, the density tends to infinity at $0$.

Discontinuity conjecture. The preceding assertion holds for every such probability measure $u$.

The converse to the sufficient condition established earlier in the paper would characterize the critical case in which the free Lévy measure has total mass one and the semicircular component vanishes. The source explicitly says that this converse remains open.