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I am trying to wrap my head around what characteristics/assumptions on average returns and volatilities of assets would make the equal risk contribution portfolio optimal in a mean-variance sense. I am trying to do this for all kinds of risk-based optimization methods (mainly: minimum variance, max diversification, and risk-parity) so that I can understand what the implied views of each technique are.

For example, the minimum variance portfolio is mean-variance optimal if average returns are independent of volatilities - this makes sense intuitively as it suggests there is little to be gained by taking on excess volatility. However, I wanted to know under what conditions an equal risk contribution portfolio is optimal, and if a proof/heuristic sketch of a proof would be available.

In general this by write-up by ReSolve Asset Management states that:

The Equal Risk Contribution portfolio will hold all assets in positive weight, and is mean-variance optimal when all assets are expected to contribute equal marginal Sharpe ratios (relative to the Equal Risk Contribution portfolio itself)

but I was wondering if anyone could provide a short proof of this statement.

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