Get current database connection in Visual Studio extension

Question

I'm trying to write an extension for Visual Studio (2019) which will add new Intellisense capabilities when writing SQL queries.

I have found an example on how to extend the text editor and show suggestions. But in order for the suggestions to be meaningful, I would like to get the connection to the database used in that query window:

And then use it to create those suggestions.

Any idea how to achieve this?