  We show that BMO-solvability implies scale invariant quantitative absolute
continuity (specifically, the weak-$A_\infty$ property) of caloric measure with
respect to surface measure, for an open set $\Omega \subset \mathbb{R}^{n+1}$,
assuming as a background hypothesis only that the essential boundary of
$\Omega$ satisfies an appropriate parabolic version of Ahlfors-David
regularity, entailing some backwards in time thickness. Since the
weak-$A_\infty$ property of the caloric measure is equivalent to $L^p$
solvability of the initial-Dirichlet problem, we may then deduce that
$BMO$-solvability implies $L^p$ solvability for some finite $p$.
