The cap-induced factorization conjecture for even-dimensional projective spaces

An $M$-cap in $\mathrm{PG}(n,q)$ is a set of $M$ points with no three collinear. By Ebert's theorem, if $n\geq 2$ is even, then $\mathrm{PG}(2n-1,q)$ can be partitioned into disjoint $M$-caps, where $M=q^n+1$. Cap-induced factorization conjecture. There exists an $(M-1)$-factorization of $\lambda K_v$, where $v=\tfrac{q^{2n}-1}{q-1}$, for some $\lambda$ induced by this set of $M$-caps, when $n\geq 2$ is even. This is a proposed construction of graph factorizations from cap partitions in finite projective spaces; the source does not provide a resolution.