Maximum $2$-scattered subspaces of $V(r,q^6)$

For every prime power $q$ and every integer $r\ge 5$ coprime to $6$, we construct a maximum $2$-scattered ${\mathbb{F}}_q$-subspace of $V(r,q^6)$. The construction is a two-dimensional ${\mathbb{F}}_{q^r}$-subspace of ${\mathbb{F}}_{q^{6r}}$, viewed as an $r$-dimensional vector space over ${\mathbb{F}}_{q^6}$. A trace argument reduces the proof to the complementarity of two ${\mathbb{F}}_{q^r}$-subspaces. We establish this complementarity by separating two cases, which lead to a cubic polynomial obstruction and a quadratic norm obstruction. The associated rank-metric code is a $[2r,r,4{]}_{q^6\mathrm{/}q}$ MRD code equivalent to its dual. Taking the cases $r=5,7$ and direct sums with the standard dimension-three construction gives maximum $2$-scattered subspaces of $V(r,q^6)$ for every $q$ and every $r\ge 3$ except $r=4$. When $q$ is an odd power of $2$, the known dimension-four construction also covers this remaining case.