  We establish a sharp Sobolev trace inequality on the Siegel domain
$\Omega_{n+1}$ involving the weighted norm-$W^{2,2}(\Omega_{n+1},
\rho^{1-2[\gamma]})$. The inequality is closely related the realization of
fractional powers of the sub-Laplacian on the Heisenberg group $H^n=\partial
\Omega_{n+1}$ as generalized Dirichlet-to-Neumann operators associated to the
weighted poly-sublaplacian, generalizing observations of
Frank--Gonz\'alez--Monticelli--Tan.
