Berkovich correspondence conjecture for deeper bubbles

Let $\pi\colon\mathcal{X}\to\Delta^{\rm conv}(\mathbb{C})$ be an affine finite-type log-terminal faithfully flat morphism, where $\Delta^{\rm conv}(\mathbb{C})=\operatorname{Spec}(\mathbb{C}[[t]]^{\rm conv})$ and $v(t)=1$. Let $\mathcal{X}_\eta^{\rm an}$ be the Berkovich analytification of the generic fiber, and let $\mathfrak{B}(x\in\mathcal{X})$ denote rescaling classes of non-flat bubbles at $x$. Deeper-bubble conjecture. There is a bijection

$$\mathfrak{B}(x\in\mathcal{X})\simeq\operatorname{Crit}(x\in\mathcal{X})\subset\mathcal{X}_\eta^{\rm an},$$

with inverse maps sending a bubble class to its valuation and a valuation to its bubble. The set $\operatorname{Crit}(x\in\mathcal{X})$ and the bubble moduli space have locally piecewise-linear structure, and both depend only on the germ of $\pi$ around $x$. This predicts that all deeper bubbles, including their algebraic structures and metrics, are encoded by a locally piecewise-linear subset of the non-archimedean analytification and are local in the degeneration germ.