This intuition is not correct. For a random point in a high-dimensional space--yes--I totally agree it's highly unlikely all the eigenvalues will have the same

This intuition is not correct. For a random point in a high-dimensional space--yes--I totally agree it's highly unlikely all the eigenvalues will have the same sign (as is needed for a local minimum). The problem with Surya's claim is that the set needs to be restricted to just the critical points (where the derivative is zero). It's very hard to describe the spectral properties generally for this set, and there has been results for only 2 and 3 dimensions as far as I'm aware. Maybe the measure is the same, at least for Deep Nets, and thus a critical point is as unlikely to be a local minima as a random point is. But no work has shown that. Furthermore, we know the number of critical points to be exponential for broad classes of functions so the subset cannot be ignored. ELI5 version: One can ask, what's the probability that a random person drawn from Earth's population (points in high dimensions) owns a Bugatti (is a local minimum)? It's very small, obviously. But that doesn't tell us anything about the probability of Bugatti ownership among select subsets of people (critical points).