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VS Code has support for it on a per-theme basis. I am currently using the Atom One Dark theme on VS Code, and Operator Mono, it does work:

Above being a random screenshot from a project laying around.

I haven't found how to persist those italic settings. I can use the inspector and set them, but as soon as I close the app I loose them.

As I mentioned, it's on a per-theme basis.

Try installing 10-15 themes, and then trying them one by one. Some will have italic comments, some won't.