### 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
ERR_FAIL_COND(p_event.is_null());
	Ref<InputEventKey> k = p_event;
	if ((k.is_valid() && k->is_pressed() && !k->is_echo()) || Object::cast_to<InputEventShortcut>(*p_event)) {
		EditorPlugin *old_editor = editor_plugin_screen;
		if (ED_IS_SHORTCUT("editor/filter_files", p_event)) {
			FileSystemDock::get_singleton()->focus_on_filter();
			get_tree()->get_root()->set_input_as_handled();
		}
		if (ED_IS_SHORTCUT("editor/editor_2d", p_event)) {
			editor_select(EDITOR_2D);
		} else if (ED_IS_SHORTCUT("editor/editor_3d", p_event)) {
			editor_select(EDITOR_3D);
		} else if (ED_IS_SHORTCUT("editor/editor_script", p_event)) {
			editor_select(EDITOR_SCRIPT);
		} else if (ED_IS_SHORTCUT("editor/editor_help", p_event)) {
			emit_signal(SNAME("request_help_search"), "");
		} else if (ED_IS_SHORTCUT("editor/editor_assetlib", p_event) && AssetLibraryEditorPlugin::is_available()) {
			editor_select(EDITOR_ASSETLIB);
		} else if (ED_IS_SHORTCUT("editor/editor_next", p_event)) {
			_editor_select_next();
		} else if (ED_IS_SHORTCUT("editor/editor_prev", p_event)) {
			_editor_select_prev();
		} else if (ED_IS_SHORTCUT("editor/command_palette", p_event)) {
			_open_command_palette();