....... He still has the cord around his neck..... 45 minutes later.....
....... left this cord around his neck the whole time?
He probably assumed it was evidence, had DNA on it and possibly fingerprints, and wanted the police to collect it from where it was left. Or, possibly, he was unaware that it was there. The fact that he left evidence in situ shouldn't be taken as meaning that it wasn't evidence, or that it was faked evidence, or that there is something otherwise fishy with his story.