For a general smooth stationary Gaussian process on the real line, can one derive the exact asymptotic joint distribution of the location of the minimum and the overshoot above a high threshold, conditioned on that minimum being unusually low?

Resolving this problem would provide a complete extreme value theory for smooth Gaussian processes analogous to the classical Fisher-Tippett-Gnedenko theorem for independent sequences, but capturing spatial information that the classical theory discards entirely. It would have immediate consequences for statistical applications including the calibration of confidence regions in Gaussian process regression, the design of global optimization algorithms that use probabilistic surrogates, and the analysis of energy landscapes in statistical physics models such as the spherical spin glass, where the geometry near the global minimum is a central quantity of interest.