how can they prove a substantial reason
You have been misled as to the requirements of SOSR, there is no requirement for BA to prove that their reason
is substantial, just that they have a genuine belief that they
have a substantial reason.
As BA have what they need at the moment in terms of savings, I would be really quite surprised if they were to use SOSR, but it is important to note that a substantial reason could quite simply be to end the dispute.