No, I really don't think they did. PR wrote the note in haste, sloppily, and it has been speculated with her left hand.
The Rs lawyers saw to it that their own handwriting experts could muddy the waters.
I truly believe that night they formed their own "short list" (though it grew to encompass nearly everyone they ever knew) of who they could pin this on. I think LHP was at the top of that list. They viewed her as expendable, someone without the resources to really fight back. They may even have hoped she would NOT get convicted- so they could feel better about blaming an innocent person. To them, she was very beneath their social standing, no matter how Patsy confided in her. She was a cleaning lady. End of story.
JR's comment to Det. Arndt IMMEDIATELY after he "found" his daughter is suspicious just by it's timing. I mean, you discover the dead body (he knew, believe me) of your 6-year old daughter, garroted and wrapped in her own blanket in your own home. And the first thing you say is "this is an inside job"? That is the kind of thing that comes out while you are trying to help LE solve the crime.
No anguished cries, no hysteria. Right from the get-go put the suspicion on the person you felt it would best stick to. The housekey, the familiarity with the home AND JB. The need for money. (guess they never thought how senseless it was for LHP to ask for ransom and not provide a victim to actually...ransom.
When the case they were building against this hapless woman never really took off (no forensic evidence, and a woman quite willing to talk to LE- she was quickly ruled out by them after willingly providing prints, hair, saliva, etc) they turned the wheels of the bus towards other "insiders".
Interesting that in the beginning the Rs themselves were not really pushing a stranger intruder. They were going for someone who was familiar. Guess they had to, with JR's comment coming so quickly after the body was found.