### 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 < extensions.size(); i++) {
				file->add_filter("*." + extensions[i], extensions[i].to_upper());
			}
			Node *scene = editor_data.get_edited_scene_root();
			if (scene) {
				file->set_current_path(scene->get_scene_file_path());
			};
			file->set_title(p_option == FILE_OPEN_SCENE ? TTR("Open Scene") : TTR("Open Base Scene"));
			file->popup_file_dialog();
		} break;
		case FILE_QUICK_OPEN: {
			quick_open->popup_dialog("Resource", true);
			quick_open->set_title(TTR("Quick Open..."));
		} break;
		case FILE_QUICK_OPEN_SCENE: {
			quick_open->popup_dialog("PackedScene", true);
			quick_open->set_title(TTR("Quick Open Scene..."));
		} break;
		case FILE_QUICK_OPEN_SCRIPT: {
			quick_open->popup_dialog("Script", true);
			quick_open->set_title(TTR("Quick Open Script..."));
		} break;
		case FILE_OPEN_PREV: {
			if (previous_scenes.is_empty()) {
				break;
			}
			opening_prev = true;
			open_request(previous_scenes.back()->get());
			previous_scenes.pop_back();
		} break;