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