Conjecture R2 on rank 2 local systems coming from abelian varieties

Let $X/\mathbb{F}_{q}$ be a smooth, geometrically connected, quasi-projective variety, let $l\neq p$ be a prime, and let $L$ be a lisse $\overline{\mathbb{Q}}_{l}$-sheaf of rank $2$ with determinant $\overline{\mathbb{Q}}_{l}(-1)$ that is irreducible with infinite geometric monodromy. Conjecture R2. There exists a non-empty open $U\subset X$ whose complement has codimension at least $2$, an abelian scheme

$$\pi\colon A_{U}\rightarrow U,$$

and natural numbers $m_{\sigma}$ such that

$$R^{1}\pi_{*}\overline{\mathbb{Q}}_{l}\cong\bigoplus_{\sigma}(^{\sigma}L)^{m_{\sigma}},$$

where $^{\sigma}L$ runs over the $l$-adic companions of $L$. This is an arithmetic analogue of the Corlette–Simpson theorem, asserting that such rank $2$ sheaves arise from families of abelian varieties; its general validity is left open.