The bounding cohomology conjecture for smooth projective surfaces

Let $X$ be a smooth projective surface, and let $C\subseteq X$ be a curve. Denote by $\mathcal O_X(C)$ the associated line bundle and by $h^i(\mathcal O_X(C))$ its cohomology dimensions.

Bounding cohomology conjecture. There exists a constant $c_X>0$ such that

$$h^1(\mathcal O_X(C))\leq c_Xh^0(\mathcal O_X(C))$$

for every curve $C$ on $X$.

This conjecture is stronger than the bounded negativity conjecture and was the main problem studied in the source. It has been disproved: a counterexample exists on a smooth projective surface with large Picard number.