The regularity conjecture for the quasi-canonical intersection space

For $0\leq r\leq n$, let $\widetilde{\mathcal{N}}^{[r]}_n$ be the formal scheme constructed from the relevant Rapoport--Zink spaces, and let $p$ be the residue characteristic. Regularity and tame normal-crossings conjecture. The formal scheme $\widetilde{\mathcal{N}}^{[r]}_n$ is regular, and its special fiber is a tame divisor with normal crossings: for every geometric point $x$, there are regular parameters $X_1,\ldots,X_n$ in the completed local ring such that

$$\varpi=\prod_{i=1}^n X_i^{m_i},$$

where each $m_i$ is prime to $p$. This local-structure conjecture would provide the regularity and controlled reduction needed for the arithmetic intersection theory developed in the paper; the general structure is described as mysterious.