Conjecture on decomposition and residual irreducibility of compatible systems

Let $K$ be a number field, and let $\mathcal{M}$ be an $n$-dimensional strictly compatible system over $K$. A compatible system is called irreducible if each representation in it is irreducible, Lie-irreducible if each representation is Lie-irreducible, and regular if its Hodge–Tate weights satisfy the regularity condition.

Compatible-systems conjecture.

1. $\mathcal{M}$ decomposes as a direct sum of irreducible strictly compatible systems.
2. If $\mathcal{M}$ is irreducible, then its representations are residually irreducible for all but finitely many primes.
3. If $\mathcal{M}$ is irreducible and regular, then it is induced from a Lie-irreducible strictly compatible system over a finite extension of $K$.

These assertions are proposed as extensions of the uniform irreducibility properties known for compatible systems arising from elliptic curves. The parser supplies no evidence that the conjecture has been resolved; its status is therefore open.