The fact that he hasn't taken a LDT means nothing to me. Keep in mind the cops are zeroing in on him from the start. If I were in his shoes, I wouldn't take the LDT either. They are useless. The only thing a LDT can do is persuade the criminals with low intellect to spill the beans, and that's all they are used for. They are in-admissable in court for a reason, and the reason is they are notoriuosly unreliable. I agree that SP is a prime suspect, as well he should be taking into consideration he was the last to see her, etc. But if I'm in his shoes, no way I take a lie detector either. How many times have you heard that a lie detector test was inconclusive? Inconclusive equals guilty, and you can bet a weeks pay that any info derived from a LDT WILL almost certainly be leaked to the press.
Like I said, he and his whereabouts need to be investigated vigorously, but all other leads must be checked out just as thoroughlly. Now, having said all of that, do you really think that if he went in and took a LDT and passed, that he would be dropped as a suspect?
I hope I am never a suspect in a crime because I don't trust LD tests and probably wouldn't take one. I would also get a lawyer even if I was innocent, if I was a suspect in a crime. All that aside, I still think there is a good chance SP is guilty.