### 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
Ref<DirAccess> dir = DirAccess::create(DirAccess::ACCESS_FILESYSTEM);
	ERR_FAIL_COND(dir.is_null());
	for (int i = 0; i < p_files.size(); i++) {
		String from = p_files[i];
		String to = to_path.path_join(from.get_file());
		if (dir->dir_exists(from)) {
			Vector<String> sub_files;
			Ref<DirAccess> sub_dir = DirAccess::open(from);
			ERR_FAIL_COND(sub_dir.is_null());
			sub_dir->list_dir_begin();
			String next_file = sub_dir->get_next();
			while (!next_file.is_empty()) {
				if (next_file == "." || next_file == "..") {
					next_file = sub_dir->get_next();
					continue;
				}
				sub_files.push_back(from.path_join(next_file));
				next_file = sub_dir->get_next();
			}
			if (!sub_files.is_empty()) {
				dir->make_dir(to);
				_add_dropped_files_recursive(sub_files, to);
			}
			continue;
		}
		dir->copy(from, to);
	}
}
void EditorNode::_file_access_close_error_notify(const String &p_str) {