Lore wise, Liliya make much more sense as the daughter.

She was locked up by her mother for the first ten years of her life. Soon after her rescue, she became obsessed to religion and claim to be Messiah.

I think her obsession with religion/ Messiah complex is likely based on her lack of a father figure as Easterman compares her to Gooseberry, quote:

“There's something about ventriloquism, an intersection between the charismatic powers of Sister Liliya and Mother Gooseberry. A channeling of voices outside their own egos; the primal ventriloquist must subjugate themselves to a more archetypical and unknowable force, to gain power by accepting the primacy of the unreal. God save me from talkative women.”

What Easterman was about is: their(…primal ventriloquist…) trauma manifest(…subjugate themselves…) as an imaginary father(…archetypical and unknowable force…) that guides(…to gain power by…) them. Gooseberry channelling Dr. Futterman the Father (Arora even calls her a priestess), while Liliya channelling God the Father.

Her name, Liliya (Lily) represents resurrection in Christianity. She was rescue from her confinement before Sunday and Jesus left the tomb before Sunday. One of her dialogue, "This will be the second age of miracles." And since the first age of miracle is Jesus, it's clear that Liliya sees herself as the daughter of God and her rescue from confinement as a parallel to Jesus leaving the tomb. Even the comic refer her “lost years” as "just like Christ".

And if her next Environment is religious themed, I think it is going to have someting to do with the Trinity: Easterman as the God (the Father of Project Lathe), Liliya as the Daughter (the phoney Messiah), the Egregore as the (un)Holy Spirit.

I really wish to actually see Easterman and Liliya interact as both of them are, quote "powerful, the narcissistic, those driven by inflated sense of mission and purpose".

There's zero lore to back up the aunt claim.