  We study the influence of phonon scattering on the single-triplet measurement
of two-electron spin states in a double quantum dot doped with two electrons.
We point out that at low temperatures phonon-induced relaxation to the ground
state suppresses transitions to doubly occupied singlet states which are the
source of detectable current fluctuations in this measurement scheme. Thus, for
a relatively strong electron-phonon interaction present in the system, the two
configurations display the same noise characteristics. In this way, coupling to
phonons reduces the distinguishability between the singlet and triplet
configurations. Under such conditions, the measurement scheme is no longer
valid even though the temporal characteristics of the measurement process and
measurement- induced localization are essentially unchanged.
