User Agent: Mozilla/5.0 (X11; Fedora; Linux x86_64; rv:66.0) Gecko/20100101 Firefox/66.0
Steps to reproduce:
In Menu->Customize... set the theme to Dark, then opened a Private Window.
The in-page search bar in the Private Window's default page appears with a white background, despite Firefox's theme having been set to Dark. This is unlike the in-page search bar on the non-private New Tab page, which is given a gray background when Firefox's theme is set to Dark.
The colour scheme of the in-page search bar in a Private Window page should follow Firefox's current theme.