Boundedness conjecture with bounded volume for klt stable minimal models

Let $d\in \mathbb{N}$, $u,v,w\in \mathbb{R}^{>0}$, and let $\Phi\subset \mathbb{R}^{\geq 0}$ be a DCC set. Consider pairs $(X,B),A\in \mathcal{S}_{klt}(d,\Phi,\leq u,v)$, where $\mathcal{S}_{klt}(d,\Phi,\leq u,v)$ is the family of klt stable minimal models with the indicated parameters. Write $\operatorname{vol}$ for the volume of a divisor.

Bounded-volume boundedness conjecture. If

$$\operatorname{vol}(K_X+B+A)<w,$$

then the set of all such $(X,B),A$ forms a bounded family.

This is presented as a weak version of the preceding boundedness conjecture. It is motivated by known results for rational coefficients, fixed fiber volume, strongly stable minimal models, and additional upper bounds on the total volume, while the stated general version remains open.