Generalized log nonvanishing and abundance conjecture for movable divisor cones

Let $(X,\Delta)$ be a projective klt pair of dimension $n$, and fix an integer $k$ with $0\leq k\leq n-1$. Let $\operatorname{Mov}^k(X)$ be the cone generated by numerical classes of effective Cartier divisors whose stable base locus has codimension at least $k+1$, and let $\overline{\operatorname{Mov}}^k(X)$ be its closure in $N^1(X)_\mathbb{R}$. Generalized log nonvanishing and abundance conjecture. If

$$K_X+\Delta\in\overline{\operatorname{Mov}}^k(X),$$

then

$$K_X+\Delta\in\operatorname{Mov}^k(X).$$

This simultaneously generalizes the log nonvanishing and log abundance conjectures, corresponding to $k=0$ and $k=n-1$, respectively. The source does not state a resolution status for this generalized conjecture.