Boundedness conjecture for klt stable minimal models

Let $d\to \mathbb{N}$, $u,v\in \mathbb{R}^{>0}$, and let $\Phi\subset \mathbb{R}^{\geq 0}$ be a DCC set. The notation $\mathcal{S}_{klt}(d,\Phi,\leq u,v)$ denotes the family of klt stable minimal models with these parameters.

Boundedness conjecture. The family $\mathcal{S}_{klt}(d,\Phi,\leq u,v)$ is bounded.

The base and general fiber are known to form bounded families under the stated hypotheses, but boundedness of the total space is posed as a natural question. Partial results include birational boundedness for rational coefficients and versions with additional hypotheses.