  A recent, integrability-based conjecture in the framework of the Wilson loop
OPE for N=4 SYM theory, predicts the leading OPE contribution for the hexagon
MHV remainder function and NMHV ratio function to all loops, in integral form.
We prove that these integrals evaluate to a particular basis of harmonic
polylogarithms, at any order in the weak coupling expansion. The proof
constitutes an algorithm for the direct computation of the integrals, which we
employ in order to obtain the full (N)MHV OPE contribution in question up to 6
loops, and certain parts of it up to 12 loops. We attach computer-readable
files with our results, as well as an algorithm implementation which may be
readily used to generate higher-loop corrections. The feasibility of obtaining
the explicit kinematical dependence of the first term in the OPE in principle
at arbitrary loop order, offers promise for the suitability of this approach as
a non-perturbative description of Wilson loops/scattering amplitudes.
