An appeal to the authority of @lakens on PubPeer (last comment of the thread). Even a one-way ANOVA is not always so simple.
https://pubpeer.com/publications/B1314681B75686D59552C160A36B82
@BorisBarbour Luckily, it is not so much an appeal to authority, as that they carefully looked at the simulations my co-authors (!) and me ran - and they rightly conclude our simulations do not cover this exact case. It's a high quality comment (in my authoritative view) ;)
I guess this is (implicitly) an example of "what biologists want"...
A test comparing means or medians without inflation of the false positive rate for groups that can be unbalanced, small (n = 3 is common), have unequal variance (some SD ratios beyond 10 in this case), non-normal and possibly from different distributions.
Loss of power would be acceptable - that would be the authors' problem. We already have enough false positives for a generation.
Let me use this case to promote Brunner Munzel Test
https://journals.sagepub.com/doi/10.1177/2515245921999602
So that paper refers to Delacre et al for the recommendation to use Welch's test for comparing means, but not in comparison to Brenner-Munzel, which they didn't test at all.
@plenartowicz @lakens
Will read with interest.