Spaces of triangularizable matrices (II): Finite fields with odd characteristic

Let $\mathbb{F}$ be a field. Denote by $t_n(\mathbb{F})$ the greatest possible dimension for a vector space of $n$-by-$n$ matrices over $\mathbb{F}$ in which every element is triangularizable over $\mathbb{F}$. It was recently proved that $t_n(\mathbb{F})=\frac{n(n+1)}{2}$ if and only if $\mathbb{F}$ is not quadratically closed. The structure of the spaces of maximal dimension was also elucidated provided $\mathbb{F}$ is infinite and not quadratically closed. In this sequel, we extend this result to finite fields with odd characteristic. More specifically, we prove that if $\mathbb{F}$ is finite with odd characteristic, then the space of all upper-triangular $n$-by-$n$ matrices is, up to conjugation, the sole vector space of $n$-by-$n$ matrices that has dimension $\frac{n(n+1)}{2}$ and consists only of triangularizable matrices.