### 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
Node *old_edited_scene_root = get_editor_data().get_edited_scene_root();
	if (old_edited_scene_root) {
		if (old_edited_scene_root->get_parent() == scene_root) {
			scene_root->remove_child(old_edited_scene_root);
		}
		old_edited_scene_root->disconnect(SNAME("replacing_by"), callable_mp(this, &EditorNode::set_edited_scene));
	}
	get_editor_data().set_edited_scene_root(p_scene);
	if (Object::cast_to<Popup>(p_scene)) {
		Object::cast_to<Popup>(p_scene)->show();
	}
	SceneTreeDock::get_singleton()->set_edited_scene(p_scene);
	if (get_tree()) {
		get_tree()->set_edited_scene_root(p_scene);
	}
	if (p_scene) {
		if (p_scene->get_parent() != scene_root) {
			scene_root->add_child(p_scene, true);
		}
		p_scene->connect(SNAME("replacing_by"), callable_mp(this, &EditorNode::set_edited_scene));
	}
}
int EditorNode::_get_current_main_editor() {