Gromov's boundedness conjecture for degree-two cohomology

Let $X$ be a closed Riemannian manifold, and let $\alpha \in H^2(X;\mathbb{R})$. The class $\alpha$ is $\widetilde{d}$-bounded if its pullback to the universal cover $\widetilde{X}$ has a bounded primitive, and it is bounded if it admits a bounded representative. Gromov's boundedness conjecture. The class $\alpha$ is $\widetilde{d}$-bounded if and only if it is bounded. This conjecture is equivalent to the exactness question for finitely presented groups in degree $2$ with real coefficients; its status is open.