In our new work, we prove that under minimal constraints on the physicality of a Wigner function, all states with purity μ<2/e satisfy the conjecture. Since we consider Wigner functions that are not necessarily physical, our results does not provide a negative answer to the conjecture. However, we find a large family of states that satisfy the conjecture, based on their purity.
Full details can be found the arXiv version of our work.