I don't have the detailed knowledge on designated Roths and 401ks to give you a definitive answer, but this reference might help you get a little further.
The book described at this site is by the most distinguished lawyer specializing in retirement planning tax laws. It is very dense, but very thorough, and very helpful if you can read statutes and regulations. It might or might not be too technical. It looks like sections 5.7 and 5.7.06, .08 would apply to your situation.
https://ataxplan.com/about-natalie-choate/
For $9 you can subscribe for a month and search the book at https://retirementbenefitsplanning.us/support/
Warning the book is aimed at lawyers and very technical and dense. But it might get you a little further. You might be able to ask the author or the e-book support for a referral. That might be the best way to get a definitive answer unless you get a better post here.
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