Thank you for the post. A couple of comments which I hope you don't mind receiving.

"What are the implications of such a conclusion? A common one is that the distribution of returns must be normal, or Gaussian." Actually no, just that the distribution must be symtetrical, not skew. There are many symmetrical distributions besides the normal distribution. You go on to talk about skewness, but this point is still important, because you could also address kurtosis.

"Annual return results were qualitatively similar across the various indices studied." These should be shown so that the reader can make this judgment call themselves.

"...we constructed monthly return distributions for 15 global stock market indices..." Did you use log returns? You should for geometric brownian motion processes. Although this will accentuate the negative skew of the distributions that are already negatively skew according to your calculations, it will reduce the positive skew of those with positive skew. This is even more important when considering longer periods like a year.

"Ten of the 15 indices exhibit left skewness, or crash risk: They are more prone to pronounced nose-dives than they are to steep upward climbs. " Are you intentionally implying that you can say something about the future based on the past performance of these indices? If so, you should make this assumption explicit.

Presumably, if you considered sub-periods, you may find that the distribution actually had positive skewness over certain periods. Skewness (ex-post) is in fact a random variable that has its own distribution, so you should expect to see positive skewness and negative skewness in a sample from a population where none exists. This is really easy to demonstrate through modelling.

"...just have too much skewness in their returns to validate the Sharpe Ratio as an appropriate measure for their risk-adjusted performance..." What is the issue with skewness in a market if you are comparing funds within that market, that all exhibit the same skewness i.e. all of the returns are biased in the same way. You could simply favour the lowest Sharpe Ratio funds, if you use Sharpe Ratio for fund selection, which seems crazy on its face but I accept some people may actually do this.