Gamma-BMT Brill–Noether inequality beyond named CY3 families

For every smooth projective threefold $(X,H)$ with $K_X\cong {\mathcal{O}}_X$, $H^1(X,{\mathcal{O}}_X)=0$, $H$ very ample, and $\mathrm{Pic}(X)=\mathbb{Z}H$, determine whether there are a smooth surface $S\in \mathrm{\mid }H\mathrm{\mid }$ and a smooth curve $C\in \mathrm{\mid }H{\mathrm{\mid }}_S\mathrm{\mid }$ such that ${\mathrm{BN}}_C<\chi ({\mathcal{O}}_X(H)),$ where ${\mathrm{BN}}_C$ and $\chi$ have the source definitions fixed on this page. The source Theorem 1.3 proves that this inequality suffices for $\mathrm{\Gamma }$-BMT with some $\mathrm{\Gamma }\cdot H\ge 0$. The open question is whether the inequality holds for every such $(X,H)$.

Compact Calabi–Yau manifold conventions

Bridgeland stability condition with support property

Gamma-BMT inequality and curve Brill–Noether number

The cited 2025 source states the inequality as Conjecture 5.9 and proves the sufficiency implication in Theorem 1.3. Its Theorem 1.4 establishes $\mathrm{\Gamma }$-BMT only for the listed weighted-projective complete intersections, Fano-fourfold divisors, and cyclic-cover families. A bounded targeted search on 2026-10-26 found no later primary source resolving Conjecture 5.9; this is not an exhaustive literature search. The conjectural inequality is not a logical premise or supplier.