Boundedness conjecture for log canonical models with fixed Iitaka volume

Fix $d\in\mathbb N$, $v\in\mathbb R_{>0}$, and a DCC set $I\subset(0,1]$. For a projective klt pair $(X,D)$ with $\dim X=d$ and $D\in I$, let $\operatorname{Ivol}(K_X+D)$ be its Iitaka volume, and let $f:X\dashrightarrow Z$ be the log canonical model of $(X,D)$. Define $\mathcal S(d,v,I)$ to be the set of such varieties $Z$ with $\operatorname{Ivol}(K_X+D)=v$. Boundedness conjecture. The set $\mathcal S(d,v,I)$ is a bounded family. This is attributed in the source to an earlier conjecture of Li; the supplied text gives no resolution, so the assertion remains open here.