Boucksom–Jonsson regularization conjecture

Let $(X,L)$ be a smooth polarized variety, and let $\mathbb{G}\subset \operatorname{Aut}^\circ(X,L)$ be a reductive complex Lie group. Let $(\mathcal{X},\mathcal{L})$ be a dominating simple normal crossing $\mathbb{G}$-equivariant big model. For a big line bundle $\mathcal{L}$, write $\operatorname{Vol}_{\mathcal{X}}(\mathcal{L})$ for its volume and $\langle \mathcal{L}^n\rangle\cdot K_{\mathcal{X}}$ for the positive-intersection product with the canonical divisor. A smooth $\mathbb{G}$-equivariant ample test configuration is a smooth ample test configuration equipped with the induced $\mathbb{G}$-action. Boucksom–Jonsson regularization conjecture. There exists a sequence of smooth $\mathbb{G}$-equivariant ample test configurations $\{(\mathcal{X}_k,\mathcal{L}_k)\}_{k\in\mathbb{N}}$ dominating $(\mathcal{X},\mathcal{L})$ such that

$$\lim_{k\to+\infty}\mathcal{L}_k^{n+1}=\operatorname{Vol}_{\mathcal{X}}(\mathcal{L}),\qquad \lim_{k\to+\infty}\mathcal{L}_k^n\cdot K_{\mathcal{X}_k}=\langle \mathcal{L}^n\rangle\cdot K_{\mathcal{X}}.$$

This regularization statement is important in the non-Archimedean approach to $K$-stability and is attributed to Boucksom and Jonsson. The source presents it as a key conjecture; no resolution is supplied here.