The bounded cohomology conjecture for smooth projective surfaces

Let $X$ be a smooth projective surface, and let $C$ be a curve on $X$. Write $h^i(\mathcal O_X(C))=\dim H^i(X,\mathcal O_X(C))$. The bounded cohomology conjecture asserts that there exists a positive constant $c_X$ such that

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

for every curve $C$ on $X$. The conjecture is stronger than the bounded negativity conjecture and is the main conjecture studied in the paper. The source states that it implies bounded negativity, while the cited work disproves it by constructing a counterexample on a surface of general type.