The Lelong-number conjecture for currents outside the $L^2_{-1}$ space

Let $X$ be a non-Kählerian surface. An exact positive $(1,1)$-current is understood in the sense used in the paper, and $L^2_{-1}(X)$ is the stated negative-order $L^2$ space. A Lelong number measures the local singularity of a positive current at a point.

Lelong-number conjecture. If $X$ admits an exact positive $(1,1)$-current not in $L^2_{-1}(X)$, then there exists on $X$ an exact positive current with a non-vanishing Lelong number at at least one point of $X$.

The conjecture connects failure of the global $L^2_{-1}$ condition with the existence of a current having positive local singularity. The next section states a refinement and proves it for all parabolic surfaces, but no general resolution is supplied here.