  In 2002, Johansson conjectured that the maximum of the Airy$_2$ process minus
the parabola $x^2$ is almost surely achieved at a unique location. This result
was proved a decade later by Corwin and Hammond; Moreno Flores, Quastel and
Remenik; and Pimentel. Up to scaling, the Airy$_2$ process minus the parabola
$x^2$ arises as the fixed time spatial marginal of the KPZ fixed point when
initialized from narrow wedge initial data. We extend this maximizer uniqueness
result to the fixed time spatial marginal of the KPZ fixed point when
initialized from any element of a very broad class of initial data.
  None of these results rules out the possibility that at random times, the KPZ
fixed point spatial marginal violates maximizer uniqueness. We prove that, for
a very broad class of initial data, it is with positive probability that the
set of such times is non-empty, and that, conditionally on this event, this set
almost surely has Hausdorff dimension two-thirds. In terms of directed
polymers, these times of maximizer non-uniqueness are instants of instability
in the zero temperature polymer measure--moments at which the endpoint jumps
from one location to another.
  Our analysis relies on the exact formula for the distribution function of the
KPZ fixed point obtained by Matetski, Quastel and Remenik, the variational
formula for the KPZ fixed point involving the Airy sheet constructed by
Dauvergne, Ortmann and Vir\'ag, and the Brownian Gibbs property for the
Airy$_2$ process minus the parabola $x^2$ demonstrated by Corwin and Hammond.
