in an ${{\mathbb F}}_q$-subspace of dimension at most one; see [@BL2000]. If we consider $V$ as an $rn$-dimensional ${{\mathbb F}}_q$-vector space, then it is well-known that the one-dimensional ${{\mathbb F}}_{q^n}$-subspaces of $V$, viewed as $n$-dimensional ${{\mathbb F}}_q$-subspaces, form an $n$-spread of $V$. This spread is called the *Desarguesian spread*. In this paper scattered will always mean scattered w.r.t. the Desarguesian spread. Blokhuis and Lavrauw [@BL2000] showed that the dimension of such subspaces is bounded by $rn/2$. After a series of papers it is now known that when $rn$ is even there always exist scattered subspaces of dimension $rn/2$; they are called *maximum scattered* [@BBL2000; @BGMP2015; @BL2000; @CSMPZ2016].

In [@CsMPZ], the authors introduced a special family of scattered subspaces, named $h$-scattered subspaces. Let $V$ be an $r$-dimensional ${{\mathbb F}}_{q^n}$-vector space and $h\leq r-1$ be a positive integer. An ${{\mathbb F}}_q$-subspace $U$ of $V$ is called $h$-*scattered* (or scattered w.r.t. the $h$-dimensional ${\mathbb{F}_{q^n}}$-subspaces) if ${\langle}U {\rangle}_{{{\mathbb F}}_{q^n}}=V$ and each $h$-dimensional ${{\mathbb F}}_{q^n}$-subspace of $V$ meets $U$ in an ${{\mathbb F}}_q$-subspace of dimension at most $h$. The $1$-scattered subspaces are the scattered subspaces generating $V$ over ${{\mathbb F}}_{q^n}$. The same definition applied to $h=r$ describes the $n$-dimensional ${{\mathbb F}}_q$-subspaces of $V$ defining canonical 