The subspace-induced factorization conjecture in finite projective spaces

Let $i$ and $n$ be such that $(i+1)\mid(n+1)$, and let the $i$-spaces of $\mathrm{PG}(n,q)$ be the relevant subspaces. Subspace-induced factorization conjecture. There exists a $(k-1)$-factorization of $\lambda_i K_v$ induced by the set of $i$-spaces in $\mathrm{PG}(n,q)$. This is stated as a proposed higher-dimensional packing construction; the supplied text does not define $k$, $v$, or $\lambda_i$ and gives no resolution.