### 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
case FILE_CLOSE_OTHERS: {
			tab_closing_menu_option = -1;
			for (int i = 0; i < editor_data.get_edited_scene_count(); i++) {
				if (i == editor_data.get_edited_scene()) {
					continue;
				}
				tabs_to_close.push_back(editor_data.get_scene_path(i));
			}
			_proceed_closing_scene_tabs();
		} break;
		case FILE_CLOSE_RIGHT: {
			tab_closing_menu_option = -1;
			for (int i = editor_data.get_edited_scene() + 1; i < editor_data.get_edited_scene_count(); i++) {
				tabs_to_close.push_back(editor_data.get_scene_path(i));
			}
			_proceed_closing_scene_tabs();
		} break;
		case FILE_CLOSE_ALL: {
			tab_closing_menu_option = -1;
			for (int i = 0; i < editor_data.get_edited_scene_count(); i++) {
				tabs_to_close.push_back(editor_data.get_scene_path(i));
			}
			_proceed_closing_scene_tabs();
		} break;
		case FILE_CLOSE: {
			_scene_tab_closed(editor_data.get_edited_scene());
		} break;
		case SCENE_TAB_CLOSE:
		case FILE_SAVE_SCENE: {
			int scene_idx = (p_option == FILE_SAVE_SCENE) ? -1 : tab_closing_idx;