I activated the option to set an open boundary for Parent's external mode (#define PARENT_EXT_BDY in agrif_oce_interp.F90). Grid connection is now entirely confined to 7 coarse grid points wide strips. The rest of the parent overlap should not matter.
I set the volume conserving option in ghosts to .false. (lk_div_cons=F in agrif_oce.F90). ssh is directly interpolated from parent in child ghosts points in that case at the expense of losing conservation. => Mattia: you should observe strong surface vertical velocities on the edges on the child domain in that case, but you should not have weird ssh values on the edges (unless you have some in the parent of course). Depending on the success of this test, set lk_div_cons=T (as before) and check the initial ssh (namelist option: nn_istate = 1 to output the initial field).