  Following a recent proposal by Cooper and Zwanziger we investigate via
$SU(2)$ lattice simulations the effect on the Coulomb gauge propagators and on
the Gribov-Zwanziger confinement mechanism of selecting the Gribov copy with
the smallest non-trivial eigenvalue of the Faddeev-Popov operator, i.e.~the one
closest to the Gribov horizon. Although such choice of gauge drives the ghost
propagator towards the prediction of continuum calculations, we find that it
actually overshoots the goal. With increasing computer time, we observe that
Gribov copies with arbitrarily small eigenvalues can be found. For such a
method to work one would therefore need further restrictions on the gauge
condition to isolate the physically relevant copies, since e.g.~the Coulomb
potential $V_C$ defined through the Faddeev-Popov operator becomes otherwise
physically meaningless. Interestingly, the Coulomb potential alternatively
defined through temporal link correlators is only marginally affected by the
smallness of the eigenvalues.
