Nonvanishing and Abundance for cones of movable divisors

Let ${\overline{\mathrm{M}\mathrm{o}\mathrm{v}}}^k(X)$ be the closure of the cone ${\mathrm{M}\mathrm{o}\mathrm{v}}^k(X)$ generated by classes of effective divisors on a projective variety $X$ with stable base locus of codimension at least $k+1$. We propose a generalized version of the Log Nonvanishing Conjecture and of the Log Abundance Conjecture for a klt pair $(X,\mathrm{\Delta })$, that is: if $K_X+\mathrm{\Delta }\in {\overline{\mathrm{M}\mathrm{o}\mathrm{v}}}^k(X)$, then $K_X+\mathrm{\Delta }\in {\mathrm{M}\mathrm{o}\mathrm{v}}^k(X)$. Moreover, we prove that if the Log Minimal Model Program, the Log Nonvanishing, and the Log Abundance hold, then so does our conjecture.