Saxl graphs of primitive affine groups with sporadic point stabilisers

Let $G$ be a permutation group on set $\Omega$. A base for is subset of $\Omega$ whose pointwise stabiliser trivial, and the size minimal cardinality base. If has $2$, then corresponding Saxl graph $\Sigma(G)$ vertex two vertices are adjacent if they form $G$. recent conjecture Burness Giudici states that finite primitive with property every have common neighbour. We investigate this when an affine point almost quasisimple unique subnormal subgroup covering sporadic simple group. verify under assumption, in all but ten cases.