The pointwise Lelong-number refinement for the $L^2_{-1}$-locus

Let $X$ be a non-Kählerian surface, and let $T$ be an exact positive $(1,1)$-current on $X$. The $L^2_{-1}$-locus $L^2_{-1}(T|X)$ is the set of points around which $T$ is locally in $L^2_{-1}$. Let $x$ be a point of $X$ and let a Lelong number be understood in the usual sense for a positive current.

Pointwise Lelong-number refinement. If $T$ is not in $L^2_{-1}(X)$ and $x otin L^2_{-1}(T|X)$, then there \exists on $X$ an exact positive $(1,1)$-current with non-vanishing Lelong number at $x$.

This strengthens the preceding conjecture by prescribing the point where the Lelong number is nonzero. The paper states that the refinement is verified for all parabolic surfaces; its status in general is not given.