About estimates of stability of contraction mappings on the first Heisenberg group in the fixed point theorem

On a symmetric $(1,q_2)$-quasimetric space $(\Bbb  H^1_{\alpha},\mathrm{Box}_{\Bbb H^1_{\alpha}}),$  where $\mathrm{Box}_{\Bbb H^1_{\alpha}}$ is the
$\mathrm{Box}$-quasimetic of the first Heisenberg group $\Bbb  H^1_{\alpha},$  we studied a constant  $\mathrm{L}_{\Phi}$  in the estimate  $\mathrm{Box}_{\Bbb H^1_{\alpha}}(u,\xi)\leq\frac{\mathrm{L}_{\Phi}\mathrm{Box}_{\Bbb H^1_{\alpha}}\big(u,\Phi(u)\big)}{1-\varepsilon}$ of stability of the $\varepsilon$-contracting mapping  $\Phi$ with respect to the  identity mapping; here $\xi$ is a fixed point of the mapping $\Phi$ and $u$ is an arbitrary point of $\Bbb  H^1_{\alpha}.$  In the paper, we got that $\mathrm{L}_{\Phi}=1$ when the mapping  $\Phi$ is the composition of the left translation and the homogeneous dilation subgroup.
Examples of the contracting mappings $\Phi$ on the first Heisenberg group such that $\mathrm{L}_{\Phi}$ is not less then  $C\sqrt{q_2}$ were found; here positive constant  $C$ does not depend on the choice of point $u\in\Bbb H^1_{\alpha}.$