Maxwell inequality for critical points and Voronoi cells

Let $V_{\alpha}(\bar x)$ be the potential of a generic configuration of point charges of the same sign, and let $a_{\alpha}^j$ denote the number of its critical points of index $j$. Let $\sharp^j$ denote the number of effective Voronoi cells of dimension $j$ in the configuration's Voronoi diagram. For an affine subspace $L$ generically intersecting that diagram, let $a_{\alpha,L}^j$ be the number of index-$j$ critical points of the restriction of $V_{\alpha}(\bar x)$ to $L$, and let $\sharp_L^j$ denote the number of Voronoi cells satisfying $\dim(S\cap L)=j$ that are effective with respect to $L$. Maxwell inequality. For every $\alpha\ge \frac{1}{2}$, one has

$$a_{\alpha}^j\le \sharp^j$$

and

$$a_{\alpha,L}^j\le \sharp_L^j.$$

The theorem preceding the conjecture establishes the corresponding one-to-one correspondence for sufficiently large $\alpha$; the conjecture asserts these upper bounds already for all $\alpha\ge \frac{1}{2}$, and its one-dimensional relative case is stated to remain open.