Existence of tight irreducible affine vector space partitions from divisible dimensions

Let $n$ be a positive integer, let $q$ be a prime power, and let $k$ be an integer with $1<k<n$ such that $k$ divides $n$. An affine vector space partition is a partition of the points of an affine space into affine subspaces; it is tight when its members have the prescribed uniform intersection properties, and irreducible when no two members lie in a common proper projective subspace. Existence conjecture. For each integer $1<k<n$ that divides $n$, there exists a tight irreducible affine vector space partition $\mathcal{U}$ of $\operatorname{PG}(n-1,q)$ of type $k^m$, where $m=q^{n-k}$. This would extend the preceding construction from the case $k=n/2$ to every proper divisor $k$ of $n$; the supplied text gives no proof or status evidence for the assertion.