The elliptic-quadric factorization conjecture in projective 3-space

An elliptic quadric is a cap in $\mathrm{PG}(3,q)$, and $\mathrm{PG}(3,q)$ can be partitioned into $q+1$ disjoint elliptic quadrics. The set of elliptic quadrics induces factors on the vertex set of $K_{q^3+q^2+q+1}$. Elliptic-quadric factorization conjecture. There exists a $q^2$-factorization of $\tfrac{1}{2}q^5(q-1)^2K_{q^3+q^2+q+1}$ induced by the set of elliptic quadrics in $\mathrm{PG}(3,q)$. This conjecture proposes a factorization arising from the incidence structure of elliptic quadrics; the source does not provide a resolution.