### 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
if (p_paths.size() > num_rows) {
		Label *label = memnew(Label);
		if (has_file && has_folder) {
			label->set_text(vformat(TTR("%d more files or folders"), p_paths.size() - num_rows));
		} else if (has_folder) {
			label->set_text(vformat(TTR("%d more folders"), p_paths.size() - num_rows));
		} else {
			label->set_text(vformat(TTR("%d more files"), p_paths.size() - num_rows));
		}
		vbox->add_child(label);
	}
	p_from->set_drag_preview(vbox); 
	Dictionary drag_data;
	drag_data["type"] = has_folder ? "files_and_dirs" : "files";
	drag_data["files"] = p_paths;
	drag_data["from"] = p_from;
	return drag_data;
}
void EditorNode::add_tool_menu_item(const String &p_name, const Callable &p_callback) {
	int idx = tool_menu->get_item_count();
	tool_menu->add_item(p_name, TOOLS_CUSTOM);
	tool_menu->set_item_metadata(idx, p_callback);
}
void EditorNode::add_tool_submenu_item(const String &p_name, PopupMenu *p_submenu) {
	ERR_FAIL_NULL(p_submenu);
	ERR_FAIL_COND(p_submenu->get_parent() != nullptr);
	tool_menu->add_child(p_submenu);
	tool_menu->add_submenu_item(p_name, p_submenu->get_name(), TOOLS_CUSTOM);
}
void EditorNode::remove_tool_menu_item(const String &p_name) {