The non-continuity conjecture for the Le Potier function on irregularity-zero surfaces

Let $(S,H)$ be a polarised surface, where $q=h^1(S,\mathcal O_S)=0$, and let $\Phi_{S,H}$ denote its Le Potier function.

Le Potier non-continuity conjecture. The function $\Phi_{S,H}$ is not continuous at $0$.

This conjecture concerns the behaviour of the Le Potier function near the origin, which is relevant to the non-emptiness of moduli spaces of semistable sheaves and their wall-crossing geometry. It is refuted: Beauville-type surfaces provide counterexamples for which the Le Potier function is continuous at $0$.