No, we just believed he had done it before, that she was not his first.
If they're gonna be arguing that he was drunk on the tape and he didn't really kill anybody else, which seems to be their theme right now, they can thrash the entire tape and say it was merely drunken bragging.
I'm unclear why he persists in pleading not guilty?
Not guilty to what exactly?
He did it.
Proves lies to FBI.
The rest?
It's likely the truth except why exactly is he placing so much value on the occult location of her remains? He was almost boasting about it, not about killing her but of his cleverness in hiding her.
As @DM said, he slept beside that murder weapon for 3 full weeks and hadn't even cleaned it correctly... if that is all accurate and he did not destroy her remains it suggests they are retrievable, at a stretch(!)