I generally do RTFM extensively before asking a question in forums 
In this case, I’ve already gone through the document you linked to multiple times, and the syntax you provide is the very first one I tried, way before posting here.
It didn’t work, and after some digging in the forums, it turned out that those are the headers that Netlify sends to the service you are proxying, not response headers for the actual end response that is sent to the visitor’s browser.
This is the post that led me to the syntax I posted (which also didn’t work): Netlify does not send custom headers in a redirect file (using https in the "to") - #15 by luke
(thanks for the fast response tho!)