Demailly's Hodge conjecture for strongly positive currents

Let $X$ be a complex variety, and let $\mathscr{T}$ be a $(p,p)$-dimensional strongly positive closed current on $X$ whose cohomology class satisfies

$$\{\mathscr{T}\} \in \mathbb{R} \otimes_{\mathbb{Z}} \big(H^{2q}(X,\mathbb{Z})/\operatorname{tors} \cap H^{q,q}(X)\big).$$

Demailly's Hodge conjecture for strongly positive currents. The current $\mathscr{T}$ is a weak limit

$$\mathscr{T}=\lim_{i \to \infty} \mathscr{T}_i, \qquad \mathscr{T}_i=\sum_j \lambda_{ij}[Z_{ij}],$$

where each $\lambda_{ij}$ is a positive real number and each $Z_{ij}$ is a $p$-dimensional subvariety of $X$. This conjecture was introduced by Demailly as a Hodge-type conjecture for positive currents, but the paper states that it is now disproven and presents counterexamples in every dimension and codimension greater than $1$.