Modularity conjecture for the rank-2 attractor at $\phi=33\pm8\sqrt{17}$

Let $\phi_{\pm}=33\pm8\sqrt{17}$, let $K=\mathbb Q(\sqrt{17})$, and let $G_K$ be the absolute Galois group of $K$. Let $H^3_{\mathrm{\acute et}}(X_{\phi_{\pm},\overline{\mathbb Q}},\mathbb Q_{\ell})$ be the associated $4$-dimensional étale cohomology representation, and let $\rho_{f,\ell}$ and $\rho_{g,\ell}$ be the representations associated with cuspidal newforms.

Modularity conjecture at $\phi_{\pm}$. The semisimplification of the representation of $G_K$ is isomorphic to the restriction to $G_K$ of

$$\rho_{f,\ell}\oplus\rho_{g,\ell}(-1),$$

where $f$ and $g$ have weights $4$ and $2$, respectively, for $\Gamma_1(34)$, character $\left(\frac{17}{\cdot}\right)$, and are labelled $\mathbf{34.4.b.a}$ and $\mathbf{34.2.b.a}$ in the LMFDB.

This is another numerical modularity prediction obtained from Frobenius factorisations in the one-parameter Calabi–Yau family. The source does not provide a proof or a resolution.