Supremum of the Airy2 process minus a parabola on a half line

Let $\aip(t)$ be the Airy$_2$ process. We show that the random variable [\sup_{t\leq\alpha}\{aip(t)-t^2}+\min{0,\alpha}^2] has the same distribution as the one-point marginal of the Airy$_{2\to1}$ process at time $\alpha$. These marginals form a family of distributions crossing over from the GUE Tracy-Widom distribution $F_{\rm GUE}(x)$ for the Gaussian Unitary Ensemble of random matrices, to a rescaled version of the GOE Tracy-Widom distribution $F_{\rm GOE}(4^{1/3}x)$ for the Gaussian Orthogonal Ensemble. Furthermore, we show that for every $\alpha$ the distribution has the same right tail decay $e^{-(4/3)x^{3/2}}$.