The hyperbolicity criterion via exact positive currents

Let $X$ be a non-Kählerian surface, and let $T$ range over its exact positive $(1,1)$-currents. Write $L^2_{-1}(X)$ for the stated negative-order $L^2$ space, and let $I(T)$ denote the linear form associated with $T$ as defined in the paper. A surface is hyperbolic when its minimal model is an Inoue surface, an Inoue–Hirzebruch surface, or an intermediate Kato surface.

Hyperbolicity criterion. If all exact positive $(1,1)$-currents $T$ on $X$ are in $L^2_{-1}(X)$ and satisfy $I(T)=0$, then $X$ is hyperbolic.

Together with the accompanying conjectures, this is intended as a partial converse to the proposition distinguishing hyperbolic and parabolic surfaces. The source recalls it from earlier work, but gives no resolution status.