The modular construction conjecture for type $A_1$ automorphic Galois representations

Let $K$ be a totally real field and let $\{\rho_{\pi, \lambda}: \mathrm{Gal}_{K} \to \mathrm{GL}_n(\overline E_{\lambda})\}_{\lambda}$ be the compatible system of $K$ defined over $E$ attached to a regular algebraic cuspidal automorphic representation $\pi$ of $\mathrm{GL}_n(\mathbb A_{K})$ satisfying conditions (a) and (b) of Theorem 1.1. After replacing $E$ by a larger field if necessary, consider strictly compatible systems of two-dimensional Hilbert modular Galois representations $\{f_{i, \lambda}: \mathrm{Gal}_{K} \to \mathrm{GL}_2(\overline{E}_{\lambda})\}_{\lambda}$, for $i=1,2,\ldots,r$, and a compatible system of one-dimensional representations $\{\chi_{\lambda}: \mathrm{Gal}_{K} \to \mathrm{GL}_1(\overline{E}_{\lambda})\}_{\lambda}$. Modular construction conjecture. There exist such systems and integers $k_1,\ldots,k_r\geq 1$ such that, for every $\lambda$,

$$\rho_{\pi, \lambda}=\operatorname{Sym}^{k_1}(f_{1, \lambda})\otimes\cdots\otimes\operatorname{Sym}^{k_r}(f_{r, \lambda})\otimes\chi_{\lambda}.$$

The surrounding text presents this as a speculation about constructing type $A_1$ automorphic Galois representations from Hilbert modular Galois representations; no resolution is supplied.