### 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
while (!eta.done.is_set()) {
		{
			MutexLock lock(eta.execute_output_mutex);
			if (prev_len != eta.output.length()) {
				String to_add = eta.output.substr(prev_len, eta.output.length());
				prev_len = eta.output.length();
				execute_outputs->add_text(to_add);
				DisplayServer::get_singleton()->process_events(); 
				Main::iteration();
			}
		}
		OS::get_singleton()->delay_usec(1000);
	}
	eta.execute_output_thread.wait_to_finish();
	execute_outputs->add_text("\nExit Code: " + itos(eta.exitcode));
	if (execute_output_dialog) {
		if (p_close_on_errors && eta.exitcode != 0) {
			execute_output_dialog->hide();
		}
		if (p_close_on_ok && eta.exitcode == 0) {
			execute_output_dialog->hide();
		}
		execute_output_dialog->get_ok_button()->set_disabled(false);
	}
	if (r_output) {
		*r_output = eta.output;
	}
	return eta.exitcode;
}
EditorNode::EditorNode() {