The reason you're seeing this behaviour is the underlying EC2 instance that runs JupyterLab gets terminated when you stop the notebook instance.
You can utilise lifecycle configuration script which is a bash script that runs when the EC2 instance spins back up, in this script, you can re-install extensions so your environment can be set-up properly when you open JupyterLab.
Below links provide examples on how to achieve this:
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