The negative-part contraction conjecture for lifted flip MMPs

Let $(X_1,B_1+M_1)$ be a projective $\mathbb{Q}$-factorial NQC dlt generalised pair such that $K_{X_1}+B_1+M_1$ is pseudoeffective. Consider a sequence of flips in a $(K_{X_1}+B_1+M_1)$-MMP:

$$(X_1,B_1+M_1)\dashrightarrow (X_2,B_2+M_2)\dashrightarrow (X_3,B_3+M_3)\dashrightarrow\dots.$$

Consider a diagram with birational morphisms $f_i\colon X_i'\to X_i$ and induced maps $\rho_i$ on top:

$$\begin{array}{ccccccccc} (X_1',B_1'+M_1')&\dashrightarrow& (X_2',B_2'+M_2')&\dashrightarrow& (X_3',B_3'+M_3')&\dashrightarrow&\dots\\ \downarrow f_1&&\downarrow f_2&&\downarrow f_3&&\\ (X_1,B_1+M_1)&\dashrightarrow& (X_2,B_2+M_2)&\dashrightarrow& (X_3,B_3+M_3)&\dashrightarrow&\dots \end{array}$$

Let $T$ be a component of $N_\sigma(K_{X'_1}+B'_1+M'_1)$ which is a log canonical centre of $(X'_1,B'_1+M'_1)$. The negative-part contraction conjecture. The MMP on top of the above diagram contracts $T$.

This is presented as an equivalent formulation of the preceding conjecture when the MMPs consist only of flips, linking termination behaviour to the Nakayama--Zariski decomposition. Its status is not resolved in the supplied source context.