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.

Uniqueness of the maximum of the Airy process minus a parabola

Johansson formulated this conjecture in 2002. It asserts that $H(t)=A(t)-t^2$ almost surely has exactly one maximizer on every interval $[-T,T]$ , and the established results also give uniqueness on $\mathbb{R}$ .

Known results

Corwin and Hammond proved uniqueness for the parabolically shifted Airy process, using absolute continuity of the Airy line ensemble with respect to Brownian motion and uniqueness of the Brownian maximum.
Moreno Flores, Quastel, and Remenik gave an independent proof while deriving the joint law of the maximum and its location.
Pimentel supplied another proof of the uniqueness statement.
Later KPZ fixed-point work confirms uniqueness at every fixed time, while distinguishing possible exceptional random times.

Current status (as of August 2026): The uniqueness conjecture is settled affirmatively, including uniqueness on $[-T,T]$ for every $T>0$ and on $\mathbb{R}$ ; no substantive objection or competing counterexample was found.

Uniqueness of the maximum of the Airy process minus a parabola

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