On the equivalence issue of a class of $2$-dimensional linear Maximum Rank Distance codes

Recently A. Neri, P. Santonastaso and F. Zullo extended a family of $2$-dimensional $\mathbb{F}_{q^{2t}}$-linear MRD codes found by G. Longobardi, G. Marino, R. Trombetti and Y. Zhou. Also, for $t \geq 5$ they determined equivalence classes of the elements in this new family, and provided the exact number of inequivalent codes in it. In this article we complete the study of the equivalence issue removing the restriction $t \geq 5$. Moreover, we prove that in the case when $t=4$, the linear sets of the projective line $\mathrm{PG}(1,q^{8})$ ensuing from codes in the relevant family, are not equivalent to any one known so far.