Unable to modify keybinding for cursor related commands of search input box in VSCode

I try to modify keybindings in VSCode and came across a problem when I try to modify cursor related commands for search input box. Following keybinding ctrl+b works well when focus is in the editor text and cursor moves to left as expected.

{
    "key": "ctrl+b",
    "command": "cursorLeft",
    "when": "editorTextFocus || searchInputBoxFocus || inSearchEditor || searchViewletFocus"
},

However when the cursor is in the search input box, keybinding doesn't have any effect, I still have to press left arrow key.

Are there any other command or specific when condition to move cursor in search input box?

Additional information for first comment:

In keyboard shortcuts troubleshooting when I press left arrow key in text editor, cursor moves left and I get following output which tells cursorLeft command is invoked:

[KeybindingService]: / Received  keydown event - modifiers: [], code: ArrowLeft, keyCode: 37, key: ArrowLeft
[KeybindingService]: | Converted keydown event - modifiers: [], code: ArrowLeft, keyCode: 15 ('LeftArrow')
[KeybindingService]: | Resolving [ArrowLeft]
[KeybindingService]: \ From 10 keybinding entries, matched cursorLeft, when: textInputFocus, source: built-in.
[KeybindingService]: + Invoking command cursorLeft.

In keyboard shortcuts troubleshooting when I press left arrow key in search input box, cursor moves left and I get following output, which does not give any associated command name:

[KeybindingService]: / Received  keydown event - modifiers: [], code: ArrowLeft, keyCode: 37, key: ArrowLeft
[KeybindingService]: | Converted keydown event - modifiers: [], code: ArrowLeft, keyCode: 15 ('LeftArrow')
[KeybindingService]: | Resolving [ArrowLeft]
[KeybindingService]: \ From 9 keybinding entries, no when clauses matched the context.

I think you have this issue: Keybindings don't respect editorTextFocus in the find widget.

It looks like some basic commands were hard-coded (at least for the FInd Widget but that probably applies to the Search Inputs as well) and so can't be modified by a user:

> I think I added that code initially, maybe more than 10 years ago :).
> Forwarding to you for your consideration. The ask here is to remove
> the hard-coded keybinding from findWidget.ts and create a command
> which would have the keybinding ctrl+down. That would allow users to
> rebind the behavior of ctrl+down.

Also see more generally, including search input Allow to configure cursor keys for input fields.

So the vscode team is aware of the issue and it is on the backlog - you could upvote those issues in the meantime.