  We analyze the effect of decoherence on the violation of the Clauser-Horne
(CH) inequality for the full electron counting statistics in a mesoscopic
multiterminal conductor. Our setup consists of an entangler that emits a flux
of entangled electrons into two conductors characterized by a scattering matrix
and subject to decoherence. Loss of phase memory is modeled phenomenologically
by introducing fictitious extra leads. The outgoing electrons are detected
using spin-sensitive electron counters. Given a certain average number of
incoming entangled electrons, the CH inequality is evaluated as a function of
the numbers of detected particles and on the various quantities characterizing
the scattering matrix. When decoherence is turned on, we show that the amount
of violation of the CH inequality is effectively reduced. Interestingly we find
that, by adjusting the parameters of the system, there exists a protected
region of $Q$ values for which violation holds for arbitrary strong
decoherence.
