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Is there concrete evidence that statistical arbitrage (historical vs. implied) on higher moments, specifically skewness and kurtosis, can be (significantly) done?

Working from this source, the author finds clear evidence on volatility arbitrage. Which is mainstream, very well understood, and heavily done (e.g. VIX trading). But, to take the discussion further, this research (Chapter 9 - Trading on Deviations of Implied and Historical Distributions) is specifically addressing this question, applied on DAX data, which is not a as liquid as the equivalent US options markets.

Therefore, my main question is: has anyone done any empirical work on this topic and could relate to it?

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I'm not sure if this would qualify as "empirical work" but you should definitely read Dynamic Hedging if you haven't already. Taleb talks a lot about this. To be more specific, have a look at page 264, the section called "Higher Moment Bets". I hope this is helpful, although I am not sure if this is what you are looking for.

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