  Let $\rho_1$ and $\rho_2$ be a pair of residual, odd, absolutely irreducible
two-dimensional Galois representations of a totally real number field $F$. In
this article we propose a conjecture asserting existence of "safe" chains of
compatible systems of Galois representations linking $\rho_1$ to $\rho_2$. Such
conjecture implies the generalized Serre's conjecture and is equivalent to
Serre's conjecture under a modular version of it. We prove a weak version of
the modular variant using the connectedness of certain Hecke algebras, and we
comment on possible applications of these results to establish some cases of
Langlands functoriality.
