Shokurov's canonical confinement conjecture for log-canonically saturated b-divisors

Let $(Y',B_{Y'})$ be a pair, and let $\mathfrak{M}(Y',B_{Y'})$ denote the set of b-free b-divisors that are log canonically saturated, meaning

$$\operatorname{Mov}\left\lceil\mathcal{M}+\mathcal{A}\right\rceil\leq\mathcal{M}.$$

A model has canonically confined singularities for this set if, for some $c>0$, each $\mathcal{M}\in\mathfrak{M}(Y',B_{Y'})$ has a member $\mathcal{M}'\in|\mathcal{M}|$ whose boundary together with $c\mathcal{M}'$ is canonical. CCS conjecture. There is a bounded family of models on which $\mathfrak{M}(Y',B_{Y'})$ has canonically confined singularities: for each $\mathcal{M}\in\mathfrak{M}(Y',B_{Y'})$, there are a crepant terminal model $(Y'_{\mathcal{M}},B_{\mathcal{M}})$ and $\mathcal{M}'\in|\mathcal{M}|$ such that

$$B_{Y_{\mathcal{M}}}+c\mathcal{M}'_{Y'_{\mathcal{M}}}$$

is canonical. Moreover, if $(Y,B_Y)/T$ is birational, the family can be taken finite; for the problem considered in the paper, it can be taken to consist of one model.