Boundedness conjecture for log Fano varieties

Fix $\varepsilon>0$. Let $(X,D)$ be a normal projective log variety such that

$$K+D\text{ is }{\mathbb Q}\text{-Cartier},$$ $$\operatorname{totaldiscr}(X,D)>-1+\varepsilon,$$

and

$$-(K_X+D)\text{ is nef and big}.$$

Boundedness conjecture for log Fano varieties. Then $(X,\operatorname{Supp}(D))$ belongs to a finite number of algebraic families. This conjecture is known to be true when $\dim(X)=2$; for $\dim(X)\geq 3$, only particular results are known, so the general assertion remains open.