  We construct initial data suitable for the Kerr stability conjecture, that
is, solutions to the constraint equations on a spacelike hypersurface with
boundary entering the black hole horizon that are arbitrarily decaying
perturbations of a Kerr initial data set. This results from a more general
perturbative construction on any asymptotically flat initial data set with the
topology of $\mathbb{R}^3\setminus\{r<1\}$ enjoying some analyticity near and
at the boundary. In particular, we design a suitable mixed boundary condition
for the elliptic operator of the conformal method in order to exclude the
Killing initial data sets (KIDS).
