Conjecture on nonsingularity of polynomially bisected directions

Fix $n$, and let $I^n\subseteq\mathbb{R}^n$ be the unit cube. Given $\varepsilon,H>0$, a positive integer $N$, and a polynomial $Q$, subdivide $I^n$ into cubes of edge length $N^{-1-\varepsilon}$. A direction $v\in S^{n-1}$ is $N,\varepsilon,H$-singular with respect to $Q$ if some line in direction $v$ intersects at least $H^{-1}N^{1+\varepsilon}$ such cubes, each of which is roughly bisected by $Q$, meaning that $Q$ divides it into two parts each having volume at least one tenth of the cube's volume. Polynomial singular-direction conjecture. For every fixed $\varepsilon,H>0$, and every polynomial $Q$ whose degree is at most a sufficiently large function of $N$, the set of $N,\varepsilon,H$-singular directions is not all of $S^{n-1}$. This conjecture is presented as slightly stronger than the Minkowski Kakeya conjecture and is intended to control the continuity obstruction in the discrete-to-continuous argument. Its resolution is not supplied in the paper.