  We show that for a uniformly elliptic divergence form operator $L$, defined
in an open set $\Omega$ with Ahlfors-David regular boundary, BMO-solvability
implies scale invariant quantitative absolute continuity (the weak-$A_\infty$
property) of elliptic-harmonic measure with respect to surface measure on
$\partial \Omega$. We do not impose any connectivity hypothesis, qualitative or
quantitative; in particular, we do not assume the Harnack Chain condition, even
within individual connected components of $\Omega$. In this generality, our
results are new even for the Laplacian. Moreover, we obtain a converse, under
the additional assumption that $\Omega$ satisfies an interior Corkscrew
condition, in the special case that $L$ is the Laplacian.
