  We derive an explicit expression for the Lellouch-L\"uscher (LL) factor in
the $K\to 3\pi$ decays at leading order (without derivative couplings). Several
important technical details are addressed, like a proper decomposition into the
isospin amplitudes, the choice of a minimal set of effective couplings and the
renormalization, as well as the algorithm for the solution of the pertinent
Faddeev equations in the infinite volume which is based on the contour
deformation method. Most importantly, our numerical results demonstrate that
the three-body force contributes very little to the LL factor. This result
paves the way for the study of the $K\to 3\pi$ decays on the lattice.
