Arhancet's Hodge–Dirac functional-calculus conjecture

Let $G$ be a Lie group of polynomial volume growth, generated by vector fields $X_1,\ldots,X_m$, and let $\slashed{D}_p$ be the $L^p$ realization of the Hodge–Dirac operator on

$$\mathrm{L}^p(G)\oplus_p\mathrm{L}^p(G,\ell^p_m).$$

For $0<\theta<\frac{\pi}{2}$, let $\Sigma_\theta^\pm$ denote a bisector of angle $\theta$. Arhancet's conjecture. If $1<p<\infty$ and $p\ne2$, then $\slashed{D}_p$ is bisectorial and admits a bounded $\mathrm{H}^\infty(\Sigma_\theta^\pm)$ functional calculus on $\mathrm{L}^p(G)\oplus_p\mathrm{L}^p(G,\ell^p_m)$ for some $0<\theta<\frac{\pi}{2}$. This is presented as a conjecture because the approach developed in the paper is not sufficient to prove it; the claim concerns functional calculus for Hodge–Dirac operators beyond the Hilbert-space case $p=2$.