The multiplier conjecture for spectral balls of positive Rockland operators

Let $\mathbb{G}$ be a graded Lie group with $\dim(\mathbb{G})\geq 2$, let $\mathcal{G}$ be a positive Rockland operator on $\mathbb{G}$, and let $\chi$ be the characteristic function of the unit interval $[0,1]$. Multiplier conjecture. For $1<p<\infty$, the operator

$$\chi(\mathcal{G}):L^p(\mathbb{G})\rightarrow L^p(\mathbb{G})$$

extends to a bounded linear operator if and only if $p=2$. This is the analogue for graded Lie groups of the multiplier problem for the ball; the corresponding assertion for the operator $\mathcal{R}=\Delta_x+\mathcal{G}$ on $\mathbb{R}^n\times\mathbb{G}$ is established in the paper, while the statement for $\mathcal{G}$ itself remains open.