Special termination near the divisorial image of the boundary floor

Let $(X,B)$ be a projective $\mathbb{Q}$-factorial effective dlt pair. The divisor $\lfloor B\rfloor$ is the integral part of the boundary, and its image under the relevant birational maps has a divisorial part. Special termination conjecture. There exists a $(K_X+B)$-MMP that terminates near the divisorial part of the image of $\lfloor B\rfloor$.

This is a technical special case of special termination used in the paper to derive abundance results for pairs with numerical dimension at most one. Its general status is not resolved in the supplied text.