Great tit nest box unhinged!

A pair of great tits have been busy building their nest in a box we had up last year (for a successful nesting) on the East side of our house just by our front door. The problem is that the hinge at the bottom has come undone, so the poor birds are busy stuffing it with nesting material and a lot keeps coming out the bottom! I'm so tempted to close it properly or put some Duct tape to close it -- but my guess is that I shouldn't go near it as they will abandon ship. Can you confirm for me what would be best to do -- it's so frustrating to watch the process

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