  It has been recently understood that the harmonic measure on the boundary $E
= \partial \Omega$ of a domain $\Omega$ in $\mathbb{R}^n$ is absolutely
continuous with respect to the Hausdorff measure $\mathcal{H}^{n - 1}$ on $E$
if and only if the boundary $E$ is rectifiable. Then, by G. David, M.
Engelstein, J. Feneuil, S. Mayboroda and other coauthors, a notion of harmonic
measure for Ahlfors-regular sets $E$ of higher codimension $n - d$ was
developed with the aid of the operator $L_\alpha = -\mbox{div} D_{\alpha}^{-n +
d + \alpha} \nabla$, where $\alpha > 0$ and $D_\alpha$ is a certain regularized
distance function to the set $E$. A program was launched to establish analogous
to the classical case equivalence between rectifiability of the
higher-codimensional set $E$ and good relations of the (new) harmonic and
Hausdorff measures. The sufficiency of rectifiability for quantitative absolute
continuity was only just obtained. For the other direction the main obstacle is
to prove that, roughly, the equation $L_\alpha D_\alpha = 0$ is true only when
the set $E$ is a hyperplane. In this paper we prove some first results which
indicate that the latter conjecture may be true. We also explain that a certain
natural strategy to tackle the problem does not work till the end.
