Main conjecture on log-concavity of local BPS invariants

Let $(C,0)$ be a reduced planar curve singularity over $\mathbf{k} = \mathbb{C}$, let $\delta$ be its $\delta$-invariant, and let $n_h(C,0)$, $0\leq h\leq\delta$, be the BPS invariants defined by the genus expansion of the Euler characteristics of punctual Hilbert schemes. Main conjecture. The sequence $n_0(C,0),\ldots,n_{\delta}(C,0)$ is log-concave with no internal zeros. In particular, it is unimodal. This predicts a strong form of regularity for the enumerative invariants of planar curve singularities; the source presents it as a conjecture and gives no resolution.