More thread creep
The suggestion was slowing down to allow the service to be completed - on SH staying airborne for another 10 mins is often an increase of 1/6th of the total flight time. I do not believe that any reduction in cost index (by the way a concept that any modern operator is aware of not just BA) will mean that staying airborne equals a lower fuel burn. Ground idle fuel flow on the engines means even if the engines are running for the same amount of time you are always better off on the ground - that's without shutting one down. On LH it will only become obvious that a slow down to facilitate poor planning by the CSD meaning they need more time to complete the cabin service will come at such a late point in the flight - probably at the 40 minute to run point that the same argument applies. We are not talking about running at a reduced cost index for many hours where a significant difference in fuel burn can be achieved.