Lario–Rio conjecture on minimal conductors of mod 3 representations

Let $\mathbf{P}\overline{\rho}: \mathrm{Gal}(\overline{\mathbf Q}/\mathbf Q) \rightarrow \mathrm{PGL}_2(\mathbf F_3)$ be an irreducible representation. Assume that $\mathbf{P}\overline{\rho}$ has a linear lifting $\overline{\rho}$ to $\mathrm{GL}_2(\mathbf F_3)$ with cyclotomic determinant. Let $N(\mathbf{P}\overline{\rho})$ denote the minimal Serre conductor among all such liftings. Lario–Rio conjecture. There is a linear lifting $\rho_{E,3}$ arising from an elliptic curve $E/\mathbf Q$ whose conductor is a power of $3$ times $N(\mathbf{P}\overline{\rho})$. The paper constructs a counterexample to this conjecture when $p=3$, so the assertion is refuted.