DCC conjecture for invariant Iitaka volumes

Let $d$ and $\kappa$ be positive integers with $\kappa<d$, and let $\Gamma\subset[0,1]$ be a DCC set. For a normal variety $X$ and an $\mathbb R$-divisor $D$, the invariant Iitaka dimension $\kappa_\iota(D)$ is defined using an $\mathbb R$-divisor in the $\mathbb R$-linear system of $D$, and the invariant Iitaka volume is denoted by $\operatorname{vol}^{\iota}_\kappa(D)$. Consider the set

$$\left\{\operatorname{vol}^{\iota}_\kappa(K_X+B)\mid (X,B)\text{ is projective lc},\ \dim X=d,\ B\in\Gamma,\ \kappa_\iota(K_X+B)=\kappa\right\}.$$

DCC of invariant Iitaka volumes. This set is a DCC set.

Invariant Iitaka volumes extend Iitaka volumes to cases where ordinary Iitaka dimension depends on rationality of the coefficients. The conjecture is confirmed in low dimensions, but remains open in general.