  Let $\mathbb{G}$ be any Carnot group. We prove that if a convolution type
singular integral associated with a $1$-dimensional Calder\'on-Zygmund kernel
is $L^2$-bounded on horizontal lines, with uniform bounds, then it is bounded
in $L^p, p \in (1,\infty),$ on any compact $C^{1,\alpha}, \alpha \in (0,1],$
regular curve in $\mathbb{G}$.
