  We examine the maximum dimension of a linear system of plane cubic curves
whose $\mathbb{F}_q$-members are all geometrically irreducible. Computational
evidence suggests that such a system has a maximum (projective) dimension of
$3$. As a step towards the conjecture, we prove that there exists a
$3$-dimensional linear system $\mathcal{L}$ with at most one geometrically
reducible $\mathbb{F}_q$-member.
