Forcing incognito mode on a reverse proxy

[Saint Michael]

I have a reverse proxy but for security reasons, I need to force the
client to work the closest to an Incognito session as possible.
I tried adding the following:

proxy_set_header Cookie "";
add_header Set-Cookie "cookie_name=; Expires=Thu, 01 Jan 1970 00:00:01 GMT;"; }

but it still does not work correctly.

Is there a way to do this?