Bounding cohomology on a smooth projective surface with Picard number 2

The following conjecture arose out of discussions between B. Harbourne, J. Ro'e, C. Cilberto and R. Miranda: for a smooth projective surface $X$ there exists a positive constant $c_X$ such that $h^1(\mathcal O_X©)\le c_X h^0(\mathcal O_X©)$foreveryprimedivisor$C$on$X$. When the Picard number $\rho (X)=2$, we prove that if either the Kodaira dimension $\kappa (X)=1$ and $X$ has a negative curve or $X$ has two negative curves, then this conjecture holds for $X$.