https://www.audubon.org/news/the-fascinating-and-complicated...
The actual sex chromosomes of the birds, and hence they're gametes, have significant differences between the two colours.
You can quibble over if this technically fits the current definition, but the original characterization is pretty far from "complete nonsense".
[0] https://www.sciencedirect.com/science/article/pii/S096098221...
This is an incorrect understanding of gametes and supergenes [0]. There are still only two gametes (only two sexes), but the two morphs (white and tan supergenes[0]) can only effectively reproduce with the same morph of the opposite sex (again, only two sexes, only two gametes between the four morphs). This means each morph only effectively breeds with 1/4 of the population, which gives the aberration of "four sexes", even though there is a small amount (around 1%) of cross-morph breeding.
The claim that this species truly has four sexes (four gametes) is unscientific nonsense.
Only two sex chromosomes, and acts "as though" there are four sexes, which means there aren't four sexes.
The paper does not make the claim that there are four independent sexes. Helmuth incorrectly reported a claim that the paper does not make.
"Squirrels fly through the air as though they are birds" Squirrels are not birds and the previous statement does not support that claim.
An editor of a science magazine should be willing and open to discuss science with scientists. Particularly if they're trying to help correct a misconception.