Saxl graphs of primitive affine groups with sporadic point stabilizers

Let G be a permutation group on a set ω. A base for G is a subset of ω whose pointwise stabilizer is trivial, and the base size of G is the minimal cardinality of a base for G. If G has base size 2, then the corresponding Saxl graph ς(G) has vertex set ω and two vertices are adjacent if and only if they form a base for G. A recent conjecture of Burness and Giudici states that if G is a finite primitive permutation group with base size 2, then ς(G) has the property that every two vertices have a common neighbour. We investigate this conjecture in the case where G is an affine group and a point stabilizer is an almost quasisimple group whose central quotient is either S or Aut(S) for some sporadic simple group S. We verify the conjecture for all but 16 of the groups G.