The global hypoellipticity conjecture for the Vladimirov sub-Laplacian

Let $\mathbb{K}$ be a non-archimedean local field with ring of integers $\mathscr{O}_\mathbb{K}$, prime ideal $\mathfrak{p}=\textbf{p}\mathscr{O}_\mathbb{K}$, and residue field $\mathbb{F}_q=\mathscr{O}_\mathbb{K}/\textbf{p}\mathscr{O}_\mathbb{K}$. Let $\mathfrak{g}=\operatorname{span}_{\mathscr{O}_\mathbb{K}}\{X_1,\ldots,X_d\}$ be a nilpotent $\mathscr{O}_\mathbb{K}$-Lie algebra, and let $\mathbb{G}$ be the exponential image of $\mathfrak{g}$, so that $\mathbb{G}$ is a compact nilpotent $\mathbb{K}$-Lie group. Let $X_1,\ldots,X_\kappa$, with $1\leq\kappa\leq d$, be a basis for $\mathfrak{g}/[\mathfrak{g},\mathfrak{g}]$, so that these elements generate $\mathfrak{g}$. The Vladimirov sub-Laplacian conjecture. For every $\alpha>0$, the Vladimirov sub-Laplacian $\mathscr{L}^{\alpha}_{\mathrm{sub}}$ is hypoelliptic on $\mathbb{G}$ and invertible on the space of mean-zero functions, where, for $f\in\mathcal{D}(\mathbb{G})$,

$$\mathscr{L}^{\alpha}_{\mathrm{sub}}f(\mathbf{x})=\sum_{k=1}^{\kappa}\partial_{X_k}^{\alpha}f(\mathbf{x}).$$

Here $\partial_{X_k}^{\alpha}$ denotes the directional Vladimirov-Taibleson operator of order $\alpha$ associated with $X_k$. The conjecture concerns global hypoellipticity and solvability of a distinguished pseudo-differential operator on compact nilpotent groups over non-archimedean local fields; the paper studies this question for the compact Engel group, but the supplied text gives no resolution of the conjecture.