Harbourne's bounded cohomology conjecture

Let $X$ be a smooth projective surface that is either rational or defined in characteristic zero. The conjecture asserts that there exists a constant $c_X$ such that every prime divisor $C$ satisfies

$$h^1(X,C)\leqslant c_Xh^0(X,C).$$

This conjecture was proposed as a variant related to SHGH, which corresponds to the stronger value $c_X=0$ for blow-ups of generic points of $\mathbb P^2$. The source itself notes that a later corollary gives a counterexample, so the conjecture is refuted.