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This claim is based on conversations with 4 consultants at these firms in North American and Europe.
I've heard two explanations for this:
80,000 Hours recommends investigating other paths before entering consulting, but if it makes sense in your situation now could be a good time to apply.
There was recently a sequence written on management consulting if you're interested in learning more.
Agree. You can also look at the effective altruism and consulting network homepage for more advice and reach out for career advice. Eac-network.com here are also more considerations whether you should enter consulting: https://forum.effectivealtruism.org/posts/AY9bWEtSgguRC9gTL/considerations-and-advice-on-entering-management-consulting