I don't think Eggleston printed anything himself. Unfortunately, a lot of the prints of his work currently being offered for sale are made in volume inkjet prints, and often in somewhat large size. This is my opinion spoils the charm of the earlier dye transfer look, being out of character.
But the notion of dye transfer being more archival is no longer valid. It all depends on the specific dyes which were employed and the storage and display conditions. Cibachrome was generally a lot more permanent, and now chromogenic prints have greatly improved in terms of permanence. Inkjet is itself a complex topic, with lots of misunderstandings or even misleading marketing claims. Inkjets are not true pigment prints, but very complex blends of micro-pigments, dyes, and dyed inert particles (lakes). So it all depends. Some museums have the ability to carefully conserve and store things away. Most dye transfer prints do quite poorly around UV light, either artificial or sunlight, or in the presence of polluted air.
I'm in contact with a number of remaining dye transfer printers, and myself have enough supplies to do maybe five years printing, but even retired haven't found blocks of free time sufficient to do much except some preliminary tests and making a certain number of sets of appropriate 8X10 separation negatives. It's a time and elbow room intensive process. I'm still buried in other darkroom commitments.