People write down passwords. We all "know" it or have done it, but since this is skeptics stackexchange, I looked for "evidence". I found a paper from 2011 where researchers watched the password creation process and user behaviour:
Of Passwords and People:
Measuring the Effect of Password-Composition Policies
Overall, 31% of participants who returned for part
two reported writing down the password they used for the study, either
electronically or on paper. (See Figure 3 for details.) We found no
significant difference in rates of reported storage on paper between
our study passwords and real email passwords; however, reported
electronic storage of study passwords is significantly lower than for
real email passwords.
I could not find a study specifically about safe owners, but I don't see why human behaviour would be different when it comes to creating and remembering a password/combination of digits, when applied in a similar context, like safes.
There is lots of anecdotal evidence about this writing down of passwords of course. I remember Richard Feynman writing about his experience as a hobbyist safe cracker when he discovered this behaviour. He would look for a written down password at the sides of the drawers.
At the places I worked, with IT people, unfortunately, writing down passwords was quite common. I saw Post Its on screens with several passwords and user names. If you asked people to not do that, those notes would not vanish but move to another "hidden" place on or under the desk.
My bet is that people anywhere at all times wrote down passwords. When a secret message travelled through the Roman Empire and arrived at a secretary, I'm pretty sure that guy had the code to decipher the message carved into the side of his clipboard.
So yes, with the research paper at hand showing that people write down passwords, I go ahead and say it is very likely that safe owners do it too, because it's the same concept.
I know this doesn't prove that they keep the note in the same room, I haven't found a study on that.