  Maximum scattered subspaces are not only objects of intrinsic interest in
finite geometry but also powerful tools for the construction of MRD-codes,
projective two-weight codes, and strongly regular graphs. In 2018 Csajb\'ok,
Marino, Polverino, and Zanella introduced a new family of maximum scattered
subspaces in $\mathbb{F}_{q^6} \times \mathbb{F}_{q^6}$ arising from
polynomials of type $f_b(x)=bx^q+x^{q^4}$ for certain choices of $b \in
\mathbb{F}_{q^6}$. Throughout characterizations for $f_{b_2}(x)$ and
$f_{b_1}(x)$ giving rise to equivalent maximum scattered subspaces, the authors
conjectured that the portion of new and inequivalent maximum scattered
subspaces obtained in this way is quite large. In this paper first we find
necessary and sufficient conditions for $b$ to obtain a maximum scattered
subspace. Such conditions were found independently with different techniques
also by Polverino and Zullo 2019. Then we prove the conjecture on the number of
new and inequivalent maximum scattered subspaces of this family.
