We consider the Kardar-Parisi-Zhang (KPZ) fixed point $\mathrm{H}(x,\tau)$
with the step initial condition and investigate the distribution of
$\mathrm{H}(x,\tau)$ conditioned on a large height at an earlier space-time
point $\mathrm{H}(x’,\tau’)$. When the height $\mathrm{H}(x’,\tau’)$ tends to
infinity, we prove that the conditional one-point distribution of
$\mathrm{H}(x,\tau)$ in the regime $\tau>\tau’$ converges to the Gaussian
Unitary Ensemble (GUE) Tracy-Widom distribution, and the next three lower order
error terms can be expressed as the derivatives of of the GUE Tracy-Widom
distribution. These KPZ-type limiting behaviors are different from the
Gaussian-type ones obtained in [Liu-Wang22] where they study the finite
dimensional distribution of $\mathrm{H}(x,\tau)$ conditioned on a large height
at a later space-time point $\mathrm{H}(x’,\tau’)$. They prove, with the step
initial condition, the conditional random field $\mathrm{H}(x,\tau)$ in the
regime $\tau<\tau’$ converges to the minimum of two independent Brownian
bridges modified by linear drifts as $\mathrm{H}(x’,\tau’)$ goes to infinity.
The two results stated above provide the phase diagram of the asymptotic
behaviors of a conditional law of KPZ fixed point in the regimes $\tau>\tau’$
and $\tau<\tau’$ when $\mathrm{H}(x’,\tau’)$ goes to infinity.
