  Bonini, Borello and Byrne started the study of saturating linear sets in
Desarguesian projective spaces, in connection with the covering problem in the
rank metric. In this paper we study \emph{$1$-saturating} linear sets in
PG$(2,q^4)$, that is $\mathbb{F}_q$-linear sets in PG$(2,q^4)$ with the
property that their secant lines cover the entire plane. By making use of a
characterization of generalized Gabidulin codes, we prove that the rank of such
a linear set is at least $5$. This answers to a recent question posed by
Bartoli, Borello and Marino.
