Generalized Lelong-number vanishing conjecture in nef codimension

Let $X$ be a compact Kähler manifold of dimension $n$, and let $k\in\mathbb{N}$ satisfy $1\leq k\leq n-1$. Let $\alpha\in H^{1,1}(X,\mathbb{R})$ be a big class. Say that $\alpha$ is nef in codimension $k$ if $\nu(\alpha,Z)=0$ for every irreducible analytic subset $Z\subset X$ of codimension at most $k$. Generalized Lelong-number vanishing conjecture. If $\alpha$ is nef in codimension $k$, then

$$\nu(\langle \alpha^{n-k}\rangle,x)=0$$

for every $x\in X$. This generalizes the preceding conjecture by imposing nefness in codimension $k$ and considering the corresponding non-pluripolar power. The source proposes it as an open extension; the case $k=1$ recovers the preceding conjecture.