Asymptotic symmetry conjecture for coefficients of cyclotomic polynomials

Let $A$ be the set of points $(c,n)$ such that $c$ is a nontrivial coefficient of $\Phi_n$, and let $A_k$ consist of those points in $A$ with $n\leq k$. For each $k$, write $[-c_k,c_k]\times[0,n_k]$ for the smallest rectangle containing $A_k$. Let $A_k^+$ and $A_k^-$ be the subsets of $A_k$ with positive and negative first coordinate, respectively. If $S$ is a finite subset of the upper half-plane, let $S^\sharp$ denote its reflection across the $n$-axis; for $c',n'>0$, let $[c',n']S=\{(c/c',n/n'):(c,n)\in S\}$, and let $\mathcal H$ be the Hausdorff distance in $[0,1]^2$. Asymptotic symmetry conjecture. For every positive integer $k$, there are subsets $L_k\subseteq A_k^+$ and $M_k\subseteq A_k^-$ such that

$$\lim_{k\to\infty}\mathcal H\left([c_k,n_k]L_k,\bigl([c_k,n_k]M_k\bigr)^\sharp\right)=0$$

and

$$\lim_{k\to\infty}\frac{|A_k^+|}{|L_k|}=\lim_{k\to\infty}\frac{|A_k^-|}{|M_k|}=1.$$

The conjecture formalizes the observed increasing symmetry of the positive and negative nontrivial coefficient plots after normalization. A similar conjecture is made for the family $\{B_k\}$ of points corresponding to all coefficients described in the paper; its precise statement is not supplied here.