The recent discussion regarding what the killer may have shown Liz, I've began thinking about someone who was having an affair with S (i know LE hasn't found any evidence of it), what if she had previously told Liz that she was with S and Liz didn't believe her. Then the killer shows Liz proof that she was with S and killed Liz. Like a "look, see I told you we were together"?
I'm just trying to think of anything that was so important for the killer to show Liz before killing her. I realize this is a reach.
One more thought, but not really my going theory. Hired killer, shows Liz a picture of herself and says "is this you?" Liz says "yes", killer shoots her.