The lifting, bounded-conductor, purity and integrality conjecture for compatible systems

Let $K$ be a number field, let $L$ be a number field, and let $\{\rho_{\wp}\}$ be a strictly compatible system as in Definition, with characteristic polynomials $f_r(X)$ and exceptional set $S$. For each prime $\wp$ of $L$ above the rational prime $p$, let $\operatorname{Nm}_{\wp}$ denote the $\wp$-adic cyclotomic character.

Galois-theoretic conjecture. The system satisfies all of the following:

• Lifting: It lifts to, that is, it is the reduction up to semisimplification of, a strictly compatible system of semisimple $\wp$-adic representations.
• Bounded conductor: The prime-to-$p$ part of the Artin conductor of $\rho_{\wp}$ is independent of $\wp$ for almost all $\wp$.
• Purity: If $\rho_{\wp}$ is irreducible for almost all $\wp$, then the roots of $f_r(X)$ for primes $r$ not in $S$ have absolute value $|\operatorname{Nm}(r)|^t$ under all embeddings of $\overline{\mathbf Q}$ into $\mathbf C$, for an integer or half-integer $t$ independent of $r$. Here $\operatorname{Nm}$ is the norm map to $\mathbf Q$.
• Integrality: The twist $\{\rho_{\wp}\otimes\operatorname{Nm}_{\wp}^m\}$ is integral for some integer $m$.

These properties are presented as the more specific, purely Galois-theoretic conjecture following the motivic meta-conjecture. The supplied text gives no evidence that they have been resolved in the stated generality.