### 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
switch (p_what) {
		case NOTIFICATION_POSTINITIALIZE: {
			EditorHelp::generate_doc();
		} break;
		case NOTIFICATION_PROCESS: {
			if (opening_prev && !confirmation->is_visible()) {
				opening_prev = false;
			}
			bool global_unsaved = EditorUndoRedoManager::get_singleton()->is_history_unsaved(EditorUndoRedoManager::GLOBAL_HISTORY);
			bool scene_or_global_unsaved = global_unsaved || EditorUndoRedoManager::get_singleton()->is_history_unsaved(editor_data.get_current_edited_scene_history_id());
			if (unsaved_cache != scene_or_global_unsaved) {
				unsaved_cache = scene_or_global_unsaved;
				_update_title();
			}
			if (editor_data.is_scene_changed(-1)) {
				scene_tabs->update_scene_tabs();
			}
			
			uint64_t frame = Engine::get_singleton()->get_frames_drawn();
			uint64_t tick = OS::get_singleton()->get_ticks_msec();
			if (frame != update_spinner_step_frame && (tick - update_spinner_step_msec) > (1000 / 8)) {
				update_spinner_step++;
				if (update_spinner_step >= 8) {
					update_spinner_step = 0;
				}
				update_spinner_step_msec = tick;