Thanks for reading and for your question.
Min instance keep an instance always warm, with all the data/variable in memory. It’s efficient when you open a connection to the database and to keep it up and running.
However, sometime, the connection can be closed by the remote host and the instance must to re-establish the connection when a request comes in: it’s possible that you see that case in your cold start with min instance.
Sharing your case with a piece of code can be interesting on Stack Overflow!