  In this work we present results on the classification of
$\mathbb{F}_{q^n}$-linear MRD codes of dimension three. In particular, using
connections with certain algebraic varieties over finite fields, we provide
non-existence results for MRD codes $\mathcal{C}=\langle x^{q^t}, F(x), G(x)
\rangle \subseteq \mathcal{L}_{n,q}$ of exceptional type, i.e. such that
$\mathcal{C}$ is MRD over infinite many extensions of the field
$\mathbb{F}_{q^n}$. These results partially address a conjecture of Bartoli,
Zini and Zullo in 2023.
