I believe that if you take a whole instance backup using Atlassians complete project export tool that this would back up the Script Listeners as the scripts for these are stored encrypted inside of Jira which means that they should be backed up with there encrypted contents.
However, I would advise you to generate a backup of your instance and to extract the generated export files to check if the Listeners are backed up to confirm if this is the case.
I would also advise as a best practice manually saving the code and configuration for your listeners to files which you can back up in a version control system so that you have a separate backup of this.
I hope this information helps.
Pre-receive hooks that verify the Git commit message, the modified files, and implement similar code change controls used to be requirements of large enterprises working in regulated industries only....
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