  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
started from narrow wedge initial data. We extend this maximizer uniqueness
result to the fixed time spatial marginal of the KPZ fixed point when begun
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. To understand this
possibility, we study the probability that the KPZ fixed point has, at a given
time, two or more locations where its value is close to the maximum, obtaining
quantitative upper and lower bounds in terms of the degree of closeness for a
very broad class of initial data. We also compute a quantity akin to the joint
density of the locations of two maximizers and the maximum value. As a
consequence, the set of times of maximizer non-uniqueness almost surely has
Hausdorff dimension at most two-thirds.
  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.
