Boundedness conjecture for singular integrals on intrinsic Lipschitz graphs in the Heisenberg group

Let $\mathbb{W} \subset \mathbb{H}$ be a vertical subgroup with complementary subgroup $\mathbb{V}$, and let $\phi \colon \mathbb{W} \to \mathbb{V}$ be an intrinsic Lipschitz function. Let $T$ be a convolution type singular integral operator with a $3$-dimensional Calderón–Zygmund kernel satisfying the uniform boundedness of the vertical boundary values condition. Let $\Gamma(\phi)$ denote the intrinsic graph of $\phi$, and let a measure be $3$-Ahlfors–David regular if it satisfies the usual two-sided $3$-dimensional ball estimates. Then the boundedness conjecture. If $\mu$ is any $3$-Ahlfors–David regular measure supported on $\Gamma(\phi)$, then $T$ is bounded on $L^{2}(\mu)$. In particular, this should hold for

$$\mu=\mathcal{H}^{3}|_{\Gamma(\phi)}.$$

The conjecture extends the preceding boundedness theorem from compactly supported $C^{1,\alpha}$ intrinsic graphs to arbitrary intrinsic Lipschitz graphs and asks for $L^{2}$ boundedness for every $3$-Ahlfors–David regular measure on the graph. The Hausdorff measure restriction is known to be $3$-Ahlfors–David regular by the cited theorem, but the conjecture itself is presented as a natural question and its resolution is not supplied here.