Boundedness conjecture for canonical models with fixed Iitaka volume

Let $n$ be a positive integer, let $v$ be a non-negative rational number, and let $cmathcal{I}\subset [0,1]\cap cmathbb{Q}$ be a DCC set of rational numbers. For a klt pair $(X,cDelta)$ of dimension $n$ with $cmathrm{coeff} cDeltasubset cmathcal{I}$ and $cmathrm{Ivol}(K_X+cDelta)=v$, let

$$Z=cmathrm{Proj}coprod_{m=0}^{cinfty}H^0(X,cmathcal{O}_X(m(K_X+cDelta)))$$

and let $f:Xcdashrightarrow Z$ be the Iitaka fibration associated with $K_X+cDelta$. Let $cmathcal{D}(n,v,cmathcal{I})$ be the set of all such varieties $Z$. Boundedness conjecture. The set $cmathcal{D}(n,v,cmathcal{I})$ is in a bounded family. The canonical ring is finitely generated, so $Z$ is well defined; the conjecture is known when $K_X+cDelta$ is big, when the general fiber is of epsilon-lc Fano type, and when the general fiber belongs to a bounded family, but remains open in general.