Extension of vanishing and positivity of Chern classes beyond small unbounded loci

Let $X$ be a compact Kähler manifold and let $chat{E}=(E,h_E)$ be a Hermitian vector bundle in the setting of the preceding corollaries. The Chern currents $c_s(chat{E})$ and their intersections with an admissible current $T$ are defined as above.

Extension conjecture. The vanishing assertion $c_s(chat{E})\cap T=0$ for $s>\operatorname{rank}E$ and the positivity assertion that $c_s(chat{E})$ is a closed positive $(s,s)$-current still hold without assuming that $chat{E}$ has small unbounded locus.

This would extend the preceding results from metrics with small unbounded locus to arbitrary metrics in the stated setting. The source does not provide evidence resolving this claim.