Existence and boundedness of n-complements

Let $X/Z\ni o$ be a pair as in the cited theorems, with special fibre $X_o$, and let $(X/Z\ni o,B+\mathcal P)$ be an Alexeev pair of index $m$ when applicable. The condition that $X/Z\ni o$ has wFt means that it has weakly finite type in the sense used in the paper.

Existence and boundedness of $n$-complements conjecture. The cited theorems should hold without assuming that $X/Z\ni o$ has wFt. The connectedness assumption on $X_o$ may instead be replaced by a bound on the number of connected components of $X_o$. The same is expected for Alexeev pairs of index $m$.

The conjecture is stated as an extension of the paper's existence and boundedness results for complements. It is open in the generality asserted here; the source notes that the conjecture holds in dimension at most $2$.