Restarting/shutting down kernel in VS Code with Jupyter extension

[invcit]

I am experiencing issues with restarting and shutting down the Wolfram Language kernel. My setup is to run VS Code (version 1.97.0) with the Jupyter notebooks extension, and Wolfram Language for Jupyter. The kernel is provided by Wolfram Engine (version 13.3).

When restarting the kernel, a new kernel is initiated, but the old one is not shut down. Additionally, closing a notebook leaves the kernel running. Even closing VS Code leaves the kernels running. This is particularly problematic, since the license that I have does not allow an arbitrary number of kernels to be started.

Using the Jupyter PowerToys extension for VS Code, one can manually control kernels by ‘interrupt’, ‘restart’, and ‘shut down’. For Wolfram Language, ‘interrupt’ shuts down the kernel, ‘restart’ simply creates a new kernel leaving the old one running, and ‘shut down’ leaves the kernel running without starting a new one.

For me, this behaviour is unique to the Wolfram Language kernel, and it does not arise, for example, when I use a Python kernel. Hence, I am posting about the issue here, even though the cause may be the complex interaction of VS Code, the Jupyter extension, and Wolfram Language for Jupyter.