Minimal-size conjecture for tight irreducible affine vector space partitions

An affine vector space partition is a partition into affine subspaces. It is tight if the intersection of the linear parts of all its pieces contains only the zero vector, and irreducible if no proper subset of more than one part has a union that is a part. The size of a partition is its number of parts.

Affine-vector-space minimal-size conjecture. The minimal size of a tight irreducible affine vector space partition of length $n$ is

$$\frac{3}{2}n-o(n).$$

Computations in the paper give minimal sizes $4,6,7,8,10$ for lengths $3,4,5,6,7$, respectively. The stated asymptotic behavior is suggested by later constructions but remains open.