Fake quadrics with bounded cohomology property

In this paper, we characterize fake quadrics $X$ satisfying the bounded cohomology property. Namely, there exists a constant $c_X>0$ such that $h^1({\mathcal{O}}_X(C))\le c_X{h}^0({\mathcal{O}}_X(C))$ for every curve $C$ on $X$.