The newness conjecture for the maximum scattered linear set $\mathcal{L}$

Let $q$ be a prime power, and let $\mathcal{L}$ denote the linear set constructed in the paper in $\operatorname{PG}(1,q^6)$. A linear set is maximum scattered if it is scattered and has maximum possible rank. Newness conjecture. The linear set $\mathcal{L}$ is a new maximum scattered linear set of $\operatorname{PG}(1,q^6)$ for each $q$ such that $q\equiv 1\pmod{4}$ and $q\not\equiv 0\pmod{5}$. This extends the computationally verified scatteredness results and the corollary establishing newness for $q\leq17$; the conjecture asserts the corresponding statement for all admissible $q$, while the general case remains open.