On sets defining few ordinary solids

Let $\mathcal{S}$ be a set of $n$ points in real four-dimensional space, no four coplanar and spanning the whole space. We prove that if the number of solids incident with exactly four points of $\mathcal{S}$ is less than $Kn^3$ for some $K=o(n^{\frac{1}{7}})$ then, for $n$ sufficiently large, all but at most $O(K)$ points of $\mathcal{S}$ are contained in the intersection of five linearly independent quadrics. Conversely, we prove that there are finite subgroups of size $n$ of an elliptic curve which span less than $\frac{1}{6}n^3$ solids containing exactly four points of $\mathcal{S}$.