Global log-concavity conjecture for BPS invariants of locally planar curves

Let $C$ be an integral projective locally planar curve over $\mathbf{k} = \mathbb{C}$, let $g=g(C)$ and $\widetilde{g}=\widetilde{g}(C)$ be its arithmetic and geometric genera, and set $\delta(C)=g-\widetilde{g}$. Let $n_h(C)$, $0\leq h\leq\delta(C)$, be the global BPS invariants defined by the genus expansion of the Euler characteristics of Hilbert schemes of points on $C$. Global BPS log-concavity conjecture. The sequence $n_0(C),\ldots,n_{\delta(C)}(C)$ is log-concave with no internal zeros. In particular, it is unimodal. This is the global analogue of the local conjecture and concerns the enumerative geometry of integral projective locally planar curves; the source provides no evidence that it has been resolved.