Hypoellipticity and invertibility of 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 $G$ is a compact nilpotent $\mathbb{K}$-Lie group. Let $X_1,\ldots,X_\kappa$, with $1\leq\kappa\leq d$, be a basis for $\mathfrak{g}/[\mathfrak{g},\mathfrak{g}]$. For $\alpha>0$, define the Vladimirov sub-Laplacian on smooth functions by

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

Vladimirov sub-Laplacian conjecture. The Vladimirov sub-Laplacian of order $\alpha>0$ is a hypoelliptic operator on $G$ and is invertible on the space of mean-zero functions.

The conjecture concerns the analytic behavior of a distinguished pseudo-differential operator associated with directions spanning the abelianization of the Lie algebra. The supplied material gives no evidence resolving it, so its status remains open.