Surprising that sources here did not quickly find a couple of online articles from 2010, on lacto-fermented mayonnaise, one being the Washington Post.
Unsure who the Facebook group is, but they are all off the mark on this one. Sometimes, a piece of information becomes dogma just because.
Foods have been fermented for centuries, are enzyme-rich in probiotics, and was an essential method for maintaining shelf life long before refrigerators were invented, and is still used in places where home cooks have no refrigeration.
Fermenting mayonnaise, although an oil-based food, is lacto-fermented with either yoghurt whey, kefir whey, or fermented juice from any properly-fermented jar of veggies (pickles, kraut, etc). Use 1 C mayo per 1 TBSP liquid whey. Once fermented, homemade mayo lasts for months in a properly-cold fridge.
Whey may be added before mayo is whipped, as opposed to after. Doing so does not produce a thin or runny mayonnaise, as some state.
Unless someone is a supertaster, no discerning tart or sour flavor is detected, as well.
Some additional information that may be of interest:
Please keep in mind that lacto-fermented mayo is not for lactose-intolerant individuals, just as using raw eggs may not be suitable for those with compromised immune systems (the Portuguese make a thick, rich milk-based mayo that has no eggs, and recipe may be found online at Leite's Culinaria).
Cooked or uncooked yolks may be lacto-fermented.
Harold McGee, noted food science author of The Curious Cook website, and book, On Food and Cooking, has a method for gently cooking egg yolks to avoid samonella, as well as American Egg Board, The Incredible Egg website.