The rational-cycle conjecture for non-Kählerian surfaces

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 indicated negative-order Sobolev space, and let $I(T)$ denote the invariant used to distinguish the two cases of exact positive currents.

Rational-cycle conjecture. If every such $T$ belongs to $L^2_{-1}(X)$, but not every such $T$ satisfies $I(T)=0$, then $X$ admits a cycle of rational curves.

This is the complementary case in the proposed classification by the regularity of exact positive currents. The surrounding discussion motivates the conjecture through the known relationship between positive currents, surface classification, and configurations of curves, but does not state a resolution.