Bounded cohomology property on a smooth projective surface with Picard number two

We say a smooth projective surface $X$ satisfies the bounded cohomology property if there exists a positive constant $c_X$ such that $h^1(\mathcal O_X©)\le c_Xh^0(\mathcal O_X©)$foreveryprimedivisor$C$on$X$. Let the closed Mori cone $\mathrm{N}\mathrm{E}(X)={\mathbb{R}}_{\ge 0}[C_1]+{\mathbb{R}}_{\ge 0}[C_2]$ such that $C_1$ and $C_2$ with $C_2^2<0$ are some curves on $X$. If either (i) the Kodaira dimension $\kappa (X)\le 1$ or (ii) $\kappa (X)=2$, the irregularity $q(X)=0$ and the Iitaka dimension $\kappa (X,C_1)=1$, then we prove that $X$ satisfies the bounded cohomology property.