I am doing a PhD in the arts & humanities from a non-target but Russel Group school in the UK. I am hoping to join MBB through an advanced degree route after I graduate. Given my background, I understand that getting a place on one of these programmes will be exceptionally hard. But referrals offer an additional problem - given my background I have no connections with anyone in consulting, and my institution only has a handful of alumni currently working at MBB on Linkedin and their names and details are not visible (just says 'linkedin member') so I cannot contact them to initiate any conversations.
I have also considered messaging MBB employees who have my disciplinary background via Linkedin, to ask about their entry experience and how they use their skills developed in our discipline as a consultant. Ultimately hoping this leads to an eventual referral. But would cold messaging an employee like this be appropriate? I have seen comments on reddit of MBB employees stating they would only give a referral to a friend or fellow classmates and it is very disheartening but partly explains why MBB is dominated by Oxbridge and LSE alumni in the UK.