Search Engines added by Extensions can't be referenced by _Keyword_ + Tab in Private Window

Given the user has installed an Extension which adds a Search Engine
And the user has modified the Extension so that "Allow in private" is true.
Given the user is in a Private Window
And the Address Bar has Focus
When the user types the Extension's Search Keyword and presses Tab
Then further input is directed to the Extension's Search Engine
Scenario fails for Private Window:

Works with a 'normal' Window:

brave://settings/searchEngines
brave://extensions/?id=hdokiejnpimakedhajhdlcegeplioahd
lp) 
Tab Cycles through the History list that appears in the Address bar. Control is not routed to the Extension's Search Engine.
Private Window should behave as a 'normal' Window does. Input should be routed to the Extension's Search Engine

Always reproduces
Brave | 1.10.97 Chromium: 83.0.4103.116聽(Official Build)聽(64-bit)
-- | --
Revision | 8f0c18b4dca9b6699eb629be0f51810c24fb6428-refs/branch-heads/4103@{#716}
OS | Windows聽10 OS Version 1909 (Build 18363.900)

Hi!..
I'm facing that too...
I've developed a Chrome extension which uses Chromium's Omnibox API for quickly search anything, anywhere. Then uploaded it to Chrome web store which I called it "smartbox: Smart Searching".
If you're interested in, you can take a look at here.
It works pretty well with Google Chrome (works on incognito mode too). On Brave, it is not working.
Please fix this issue. Maybe this is not an issue but. Make an exception for this to work on both private and Tor sessions too.
Have a great day.
Related (or possible duplicate of) https://github.com/brave/brave-browser/issues/10549
Agree this and #10549 appear to be the same root issue. But I will point out, mine has more pictures :D
Most helpful comment
Agree this and #10549 appear to be the same root issue. But I will point out, mine has more pictures :D