The cyclotomic-divisibility equidistribution conjecture for Fekete polynomials

Let $n$ be a positive squarefree integer, let $d>1$ be an integer not dividing $n$, set $d_1=\gcd(d,n)$, and let

$$S=\\{1\leq a\leq d\mid \gcd(a,d_1)=1\\}.$$

Let $F_n$ be the Fekete polynomial and let $F_S(x)=\sum_{s\in S}x^s$. The cyclotomic-divisibility equidistribution conjecture. The elements of $\\{1\leq a\leq n\mid \gcd(a,n)=1\\}$, reduced modulo $d$, equidistribute in $S$ if and only if the $d$-th cyclotomic polynomial $\Phi_d$ divides $F_n$. Equivalently,

$$x^d-1\mid F_n-\frac{\varphi(n)d_1}{\varphi(d_1)d}F_S.$$

The proposition preceding this conjecture proves the equidistribution condition as a sufficient condition for cyclotomic divisibility, and the converse holds in all computed cases and in the cases covered by the cited theorem; its validity in general remains open.