The generic-metric bound on intersecting mediatrices

Let $(X,\tilde g)$ be the universal cover of a $d$-dimensional smooth Riemannian manifold $(M,g)$. A mediatrix is the locus equidistant from the reference point and one of its nontrivial translates under the deck-transformation group. For a generic metric $g$ on $M$, the mediatrix intersection conjecture. No more than $d$ mediatrices intersect in any given point $y$ of $X$. This predicts a general-position bound on the number of geodesics focusing at a point in the universal cover; the supplied text gives no resolution, so the conjecture remains open.