It gets worse when you realize it would’ve been impossible for the victim to put her signature on the perp’s clothes while being strangulated. Writing blind is already hard enough as it is. In this case, both their bodies would’ve been moving a lot during the commotion. Are we supposed to believe the killer was perfectly still while strangling the victim? Or that he wouldn’t notice she wasn’t struggling because she needed to write something? At best, she could’ve left a lipstick smudge on his clothes, but nothing that could realistically be read or recognized as a signature.
Not to mention the culprit should’ve noticed that the victim was only using one hand to try and break free. Then he would’ve noticed the lipstick lying around while packing her up. If she had thrown the lipstick away, it would’ve made a sound, and he would’ve checked for possible witnesses. Given how cautious he’s been so far, he would’ve wondered why she pulled out a lipstick in the first place, and checked if he had any stains anywhere on his body.