The Luxair lounge being a privately-operated lounge, I would expect that Lufthansa has arrangements for their passengers inside.
The Star Alliance lounge policy means you won't have access to the Luxair one unless you are flying on Lufthansa :
At airports where neither a Star Alliance branded lounge nor a Star Alliance member carrier offers a lounge, third party lounges are contracted by some of our member airlines. As a Star Alliance Gold customer travelling on a Star Alliance member airlines operated flight from such airports, you may have access to these third party contract lounges. Please refer to the Lounge Finder to identify which lounges you may have access to, according to the policy of each airline*.
*Check the current policy of the airline with which you are travelling.
In order to gain access to such lounges:
- You will need to show your boarding pass for a flight operated by the Star Alliance member airline that has contracted that lounge.
- If your boarding pass does not show the Star Alliance Gold indicator, you will need to also show a valid Star Alliance Gold card.
- You can bring one guest, who must be travelling on the same Star Alliance flight as you.
As an International and Domestic First Class Customer you can access
third party contract lounges contracted by the Star Alliance member
carrier on which you are travelling and located at the airport from
where your flight departs. Please present your First Class boarding
pass for a Star Alliance flight operated by a member carrier that has
a contract with the lounge. You can bring one guest who must be
departing on the same Star Alliance flight as you.
Paid Lounge Membership Customers are not entitled to use contract
lounges.