### Tested versions

v4.2.stable.official [46dc27791]

### System information

Windows 11- Godot v4.2.stable.official [46dc27791] - Vulkan (Forward+)

### Issue description

When I switch picker mode the next time I open it should be the pick mode I selected last time but it's not

https://github.com/godotengine/godot/assets/46812028/522c8b55-14cf-4781-963f-a465a03c9432



### Steps to reproduce

Toggle a picker mode off and on again

### Minimal reproduction project (MRP)

N/A
} break;
		case EditorSettings::NOTIFICATION_EDITOR_SETTINGS_CHANGED: {
			FileDialog::set_default_show_hidden_files(EDITOR_GET("filesystem/file_dialog/show_hidden_files"));
			EditorFileDialog::set_default_show_hidden_files(EDITOR_GET("filesystem/file_dialog/show_hidden_files"));
			EditorFileDialog::set_default_display_mode((EditorFileDialog::DisplayMode)EDITOR_GET("filesystem/file_dialog/display_mode").operator int());
			bool theme_changed =
					EditorSettings::get_singleton()->check_changed_settings_in_group("interface/theme") ||
					EditorSettings::get_singleton()->check_changed_settings_in_group("interface/editor/font") ||
					EditorSettings::get_singleton()->check_changed_settings_in_group("interface/editor/main_font") ||
					EditorSettings::get_singleton()->check_changed_settings_in_group("interface/editor/code_font") ||
					EditorSettings::get_singleton()->check_changed_settings_in_group("text_editor/theme") ||
					EditorSettings::get_singleton()->check_changed_settings_in_group("text_editor/help/help") ||
					EditorSettings::get_singleton()->check_changed_settings_in_group("filesystem/file_dialog/thumbnail_size") ||
					EditorSettings::get_singleton()->check_changed_settings_in_group("run/output/font_size") ||
					EditorSettings::get_singleton()->check_changed_settings_in_group("interface/touchscreen/increase_scrollbar_touch_area") ||
					EditorSettings::get_singleton()->check_changed_settings_in_group("interface/touchscreen/scale_gizmo_handles");
			if (theme_changed) {
				_update_theme();
			}
			scene_tabs->update_scene_tabs();
			recent_scenes->reset_size();
			_build_icon_type_cache();
			HashSet<String> updated_textfile_extensions;
			bool extensions_match = true;
			const Vector<String> textfile_ext = ((String)(EDITOR_GET("docks/filesystem/textfile_extensions"))).split(",", false);
			for (const String &E : textfile_ext) {
				updated_textfile_extensions.insert(E);
				if (extensions_match && !textfile_extensions.has(E)) {
					extensions_match = false;
				}