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The proof is hard to read for non-statistician. Is anyone willing to give a short human-readable summary of how it works?

Also, the conjecture seems fairly simple, so it'd be great to understand what made it hard to prove.

Here is a blog post that I think clarifies why this conjecture had remained unsolved: https://almostsure.wordpress.com/2009/09/27/the-gaussian-cor...

There is also a 'companion' article on arxiv that provides a clear presentation of Royen's proof: https://arxiv.org/abs/1512.08776