Hörmander–Mikhlin conjecture for stratified Lie groups

Let $\mathrm{G}$ be an $n$-dimensional stratified Lie group with homogeneous dilation weights $(\ell_1,\ell_2,\ldots,\ell_n)$, rescaled so that $\min(\ell_j)=1$, and let $L_\mathrm{SR}:\mathrm{G}\to\mathbf{R}_+$ be the associated subRiemannian length. For a multi-index $\gamma$, write $\{\gamma\}=\sum_{j=1}^n\ell_j\gamma_j$, and let $d_g^\gamma$ denote the corresponding derivative of a Fourier symbol $m:\mathrm{G}\to\mathbf{C}$. HM for stratified Lie groups. For every $1<p<\infty$, the Fourier multiplier $T_m$ satisfies

$$\big\|T_m:L_p(\mathcal{L}(G))\to L_p(\mathcal{L}(G))\big\|_{\mathrm{cb}}\lesssim\frac{p^2}{p-1}\sum_{|\gamma|\leq\left[\frac n2\right]+1}\big\|L_\mathrm{SR}(g)^{\{\gamma\}}d_g^\gamma m(g)\big\|_\infty.$$

This is a proposed stratified analogue of the anisotropic Hörmander–Mikhlin theorem, extending the Euclidean and anisotropic multiplier criteria to Fourier multipliers on stratified Lie groups; the parser supplies no evidence resolving it.