Is there an API to add an item to the gutters context menu?

I am developing a debugging extension which introduces a new log point type. For the best UX, I would love to provide access to the new log point the same way as vscode provides its own: Via the gutter context menu.

Is there an official way to extend the items provided in this menu?

There is gutterIconPath of DecorationRenderOptions, which allows adding a custom icon to the gutter via a FileDecorationProvider. This is a visual modification only though; it does not allow to attach any interaction to the icon.

Thanks for any pointers and suggestions :-)