  We study the evolution of axially-symmetric scalar field perturbations on an
extreme Kerr spacetime for initial data with multipole moments $\ell^{\prime}$
higher than the least radiative mode, and we measure modes $\ell$ -- and for
the first time also horizon charges -- that are excited by mode coupling
interactions. We then find the Ori-Sela prefactors, a certain quantity that can
be evaluated at finite distances and the Aretakis constant along the event
horizon of the extreme Kerr black hole for a sequence of initial data
preparations that differ only by their distance from the event horizon. We find
that for initial data in the near field there is a linear relationship of the
Aretakis constant and the Ori-Sela prefactor. For initial data farther than
these the linear relationship is not universal, and we propose that stronger
numerical simulations would be needed to regain linearity. The linear
relationship suggests that the Aretakis charge along the event horizon can be
measured at a finite distance, thereby extending this type of violation of the
no-hair theorems from the least radiative axisymmetric mode also to situations
that involve mode coupling.
