I want to set different default search engines for private and 'normal' browsing modes in Firefox, e.g use Google in normal mode, and Bing in private mode. I'm using Firefox 66.0.2 on Ubuntu 16.04, if this is relevant.
If there is no native solution, is there an existing one using something like Tampermonkey? (I know only how to use Tampermonkey-like extensions, not writing useful scripts for them...)
There is an existing similar question about Chrome, which has no native solution.