Uniqueness of the maximum of the Airy process minus a parabola

Let $A(t)$ be the Airy process and define

$$H(t)=A(t)-t^2.$$

Uniqueness conjecture. For each $T>0$, $H(t)$ has a unique point of maximum in $[-T,T]$ almost surely.

This assumption is used to establish convergence of the first maximizer $K_N$ of the discrete process to the corresponding maximizer of the limiting process. The source describes it as a very plausible assumption, but provides no proof or resolution.