What we could do with is someone from MSM interviewing the landlady and finding out exactly what it was that made her alert the police.It must have been something important for the landlady to alert the police on the same day of a disappearance without reason and more so for the police to start looking almost immediately (which lays solely with what they left behind on purpose or not). When did the police start looking? The appeal was sent out January 9th and they went missing January 7th...In the UK isn't it 24/48/72 hours at least before police start an adult missing person search unless person(s) are vulnerable and/or other mitigating circumstances.
So - we have a note, a will, keys to the flat is a possibility as @Skigh implied. Maybe locking the door then putting both keys through letterbox OR more worrying leaving the door unlocked and leaving the keys in the house. I do think the police are treading carefully here due to past cases involving MH. So a note could be the most likely explanation as to why the police seem certain on their hypothesis, for now.