One thing that confused me: had COLUMBO already solved the murder BEFORE he told Abigail she couldn't travel?..Upon subsequent viewings, I now believe he hadn't. He figured it out in real time as she watched.
He knew she was the murderer, but he needed further evidence. So, in a way it was solved, but not proved. When he smoked that cigar in the chair and the safe, he arrived at what he said to Abigail later – that Edmund wanted to reveal who killed him, but not outright, in case the killer was the one who would open the safe. He also had a sense of what the missing paper said, based in part on his conversation with Veronica. It seems that when he called on Abigail for help in solving it, her way off-base conjectures/ shenanigans came across as, once again, quite contrived… championship-level playing stupid, that further confirmed he was not only on the right track but closing in. But when he had her read what was discovered, her cover was blown away big time – and yes, in real time.