Fixes T6864. This creates a sort of busy menu but I think that's proably fine -- users are opting into activating these fields for search anyway.
In the future, we could refine this as, e.g.:
- don't show these options in the dropdown;
- do show them on some new "http prefilling" sort of page;
- then you access them as an advanced user with ?order=secret-magic.
But I'm not going to bother for now.