Adic and differential structure conjecture for potentially unramified towers

Let $L$ be a $p$-adic field, let $X/L$ be a smooth rigid analytic variety, and let $\tilde{X}/X$ be a potentially unramified tower. For each point $\tilde{x}\in|\tilde{X}|$, let $v_{\tilde{x}}$ be the associated valuation, and write

$$\hat{\tilde{X}}:=\bigl(|\tilde{X}|,\hat{\mathcal{O}}|_{|\tilde{X}|},(v_{\tilde{x}})_{\tilde{x}\in|\tilde{X}|}\bigr).$$

Potentially unramified tower conjecture. The following assertions hold: (1) $\hat{\tilde{X}}$ is an adic space and $(\hat{\tilde{X}})^\diamond=\tilde{X}^\diamond$; (2) the natural map $\mathcal{O}\mathbb{B}^+_{\mathrm{dR}}|_{|\tilde{X}|}\to\hat{\mathcal{O}}|_{|\tilde{X}|}$ is an isomorphism; (3) if $f:\hat{\tilde{X}}\to X$ is the induced map, then the resulting derivation restricts on every affinoid $\operatorname{Spa}(R,R^+)$ to the universal continuous derivation over $L$ from $R$ to an $L$-Banach module; and (4) for every connected linear algebraic group $G/L$ and conjugacy class $[\mu]$ of cocharacters of $G_{\overline{L}}$ defined over $L$, the Bialynicki–Birula map induces the stated bijection between $\operatorname{Gr}_{[\mu]}(\tilde{X}^\diamond)$ and filtrations satisfying Griffiths transversality. This predicts that potentially unramified towers over smooth rigid analytic varieties carry adic spaces with differential properties analogous to smooth rigid spaces. The source provides no resolution status.