  The fundamental problem of phase saturation of electrons in a disordered
mesoscopic system at very low temperatures is addressed. The disorder in the
medium has both static and dynamic components, the latter being in the form of
two-level systems (TLSs), which becomes just about the only source of inelastic
scattering in the limit $T\to 0$. We propose that besides the inelastic nature
of scattering from the TLSs, the phase-shift of the electrons is also affected
by the nature of tunneling in the TLSs. The tunneling becomes incoherent as $T$
decreases due to increasing long-range interactions among TLSs and affects the
phase coherence of electrons scattering from them. The competition between this
effect, which increases as $\sim T^{-1}$, and that of the scattering rate
$\tau_{e-TLS}^{-1}$ behaving as $\sim T$ apparently governs the phase-shift of
electrons.
