  Universal to black hole X-ray binaries, the high-frequency soft lag gets
longer during the hard-to-intermediate state transition, evolving from
${\lesssim}1~{\rm ms}$ to ${\sim}10~{\rm ms}$. The soft lag production
mechanism is thermal disk reprocessing of non-thermal coronal irradiation.
X-ray reverberation models account for the light-travel time delay external to
the disk, but assume instantaneous reprocessing of the irradiation inside the
electron scattering-dominated disk atmosphere. We model this neglected
scattering time delay as a random walk within an $\alpha$-disk atmosphere, with
approximate opacities. To explain soft lag trends, we consider a limiting case
of the scattering time delay that we dub the thermalization time delay, $t_{\rm
th}$; this is the time for irradiation to scatter its way down to the effective
photosphere, where it gets thermalized, and then scatter its way back out. We
demonstrate that $t_{\rm th}$ plausibly evolves from being inconsequential for
low mass accretion rates $\dot{m}$ characteristic of the hard state, to
rivaling or exceeding the light-travel time delay for $\dot{m}$ characteristic
of the intermediate state. However, our crude model confines $t_{\rm th}$ to a
narrow annulus near peak accretion power dissipation, so cannot yet explain in
detail the anomalously long-duration soft lags associated with larger disk
radii. We call for time-dependent models with accurate opacities to assess the
potential relevance of a scattering delay.
