I would assume Lillian. (Although Lily or Lilly both look correct to me, I don't know if you spell the nn one way vs. the other depending on if there is one or two Ls in the name?)
p.s. do I win a prize if you name your baby Lillian?
Probably at the very least I will reveal her name to the board as I promised. We haven't decided for sure to go with Lillian, but that's the #1 choice at the moment.
Fun fact, my top girl name is my great aunt's name (Sylvia) and Lillian is another of my great aunts! Clearly I have a type. (Other possibilities from the same generation: Daisy, Iris, Betty.)
My grandmother had a cat named Sylvia, the cat looked like Sylvester the cat (who was always chasing tweety) but was a girl, hence Sylvia.