Alexeev–Borisov boundedness conjecture for epsilon-lc weak log Fano varieties

Let $(X,B)$ be a $b4$-lc weak log Fano pair of dimension $d$, with $B$ a boundary whose coefficients lie in $91b [0,1]$. A collection of varieties is in an algebraic family when it is parametrized by finitely many algebraic families.

Alexeev\bBorisov boundedness conjecture. For every $b4>0$ and $91bb[0,1]$, the varieties $X$ for which such a pair $(X,B)$ exists are elements of an algebraic family.

The conjecture is known for surfaces, for terminal threefold singularities, for certain three-dimensional cases, and for smooth varieties in every dimension, but remains open even in dimension $3$ when $b4<1$.