Nonvanishing and Abundance for cones of movable divisors

Let Mov̄ᵏ(X) be the closure of the cone Movᵏ(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,Δ), that is: if KX+Δ ∈ Mov̄ᵏ(X), then KX+Δ ∈ Movᵏ(X). Moreover, we prove that if the Log Minimal Model Program, the Log Nonvanishing, and the Log Abundance hold, then so does our conjecture.