### 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
for (int i = 0; i < editor_data.get_edited_scene_count(); i++) {
		if (editor_data.get_scene_path(i) != p_instance_path) {
			Node *edited_scene_root = editor_data.get_edited_scene_root(i);
			if (edited_scene_root) {
				List<Node *> valid_nodes;
				find_all_instances_inheriting_path_in_node(edited_scene_root, edited_scene_root, p_instance_path, valid_nodes);
				if (valid_nodes.size() > 0) {
					edited_scene_map[i] = valid_nodes;
				}
			}
		}
	}
	if (edited_scene_map.size() > 0) {
		
		Error err;
		Ref<PackedScene> instance_scene_packed_scene = ResourceLoader::load(p_instance_path, "", ResourceFormatLoader::CACHE_MODE_REPLACE, &err);
		ERR_FAIL_COND(err != OK);
		ERR_FAIL_COND(instance_scene_packed_scene.is_null());
		HashMap<String, Ref<PackedScene>> local_scene_cache;
		local_scene_cache[p_instance_path] = instance_scene_packed_scene;
		for (const KeyValue<int, List<Node *>> &edited_scene_map_elem : edited_scene_map) {
			
			int current_scene_idx = edited_scene_map_elem.key;
			editor_data.set_edited_scene(current_scene_idx);
			Node *current_edited_scene = editor_data.get_edited_scene_root(current_scene_idx);
			
			EditorUndoRedoManager::get_singleton()->clear_history();
			
			editor_data.is_scene_changed(current_scene_idx);
			for (Node *original_node : edited_scene_map_elem.value) {