The Poisson–uncertainty conjecture for the Poisson-bracket function

Let $(M,\omega)$ be a compact symplectic manifold. For each ball capacity $c$, let $\operatorname{pb}(c)$ denote the Poisson-bracket invariant associated with the corresponding continuous covers, and call a critical value of $c$ at which the homotopy type of the space of symplectic ball embeddings changes an uncertainty phase transition. The Poisson–uncertainty conjecture. The function $\operatorname{pb}(c)$ is locally constant, upper semi-continuous and left-continuous, with jumps downwards at the uncertainty phase transitions and only at these values. The conjecture is explicitly presented as bold, and the paper later gives reasons to think it is false as stated.