Existence of a cohomologous k-positive form satisfying the subsolution inequality

Let $M$ be a compact Hermitian manifold of complex dimension $n$, let

$$be a fixed Hermitian form, and let$$

be a closed $k$-positive form. Let $c$ be the constant appearing in the equation and let $p$ range over the integers $n-l,\ldots,n-1$. A subvariety $V ewline subseteq M$ of dimension $p$ is integrated against the indicated $(p,p)$-form.

Geometric subsolution conjecture. There exists a $k$-positive form $'\text{ in }[]$ satisfying the inequality from the paper with $'$ in place of

if and only if, for every subvariety $V ewline \subseteq M$ of dimension $p=n-l,\text{... }n-1$,

\int_V c\frac{k!}{(k-n+p)!} \chi^{k-n+p}\wedge \alpha^{n-k}-\frac{l!}{(l-n+p)!} \chi^{l-n+p}\wedge \alpha^{n-l}>0.

$$This is proposed as a geometric criterion ensuring the existence of a$$

-subsolution, in analogy with the cited conjecture for the case $k=n$, $l=n-1$; its resolution is not established in the supplied text.