Hi,
Not sure whether it's the right forum but I didn't find a more relevant one.

I was flying out with LH from FCO, and I started moving from the lounge as soon as the "Boarding" announcement went up. I arrived (walking quickly, but it was far) to the gate, but it was closed. The announcement board was still saying "boarding" (not "last call").
As a result I missed my flight and my connection. Lufthansa asks me a huge amount to rebook for next day. (They would even not downgrade from Premium Econ down to Econ to fly the same day in another star alliance flight.)
Is there anything I can do, or is it entirely my fault and as a result not?
Thanks!