### 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
show_accept(
							vformat(TTR("The current scene has no root node, but %d modified external resource(s) were saved anyway."), saved),
							TTR("OK"));
				} else if (p_option == FILE_SAVE_AS_SCENE) {
					
					show_accept(
							TTR("A root node is required to save the scene. You can add a root node using the Scene tree dock."),
							TTR("OK"));
				}
				break;
			}
			file->set_file_mode(EditorFileDialog::FILE_MODE_SAVE_FILE);
			List<String> extensions;
			Ref<PackedScene> sd = memnew(PackedScene);
			ResourceSaver::get_recognized_extensions(sd, &extensions);
			file->clear_filters();
			for (int i = 0; i < extensions.size(); i++) {
				file->add_filter("*." + extensions[i], extensions[i].to_upper());
			}
			if (!scene->get_scene_file_path().is_empty()) {
				String path = scene->get_scene_file_path();
				file->set_current_path(path);
				if (extensions.size()) {
					String ext = path.get_extension().to_lower();
					if (extensions.find(ext) == nullptr) {
						file->set_current_path(path.replacen("." + ext, "." + extensions.front()->get()));
					}
				}
			} else if (extensions.size()) {
				String root_name = scene->get_name();