From the actual paper [0]:
"Our long-term genotypic analysis builds on previous work [6, 7] and, through extensive genotyping of thousands of individuals over more than two decades, confirms that white morphs are almost always heterozygous for alternative chromosome 2 alleles (2m/2). We find that 99.7% of white morphs are heterozygous (n = 1,014; Table S1) ...
As a consequence of obligate disassortative mating the species effectively has four sexes, wherein any individual can mate with only 1/4 of the individuals in the population."
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...