Möller's conjecture on scaled averages of cyclotomic coefficients

Let $a_n(k)$ denote the coefficient of $x^k$ in the cyclotomic polynomial of order $n$, and let $M_n(a_n(k))$ denote the limiting mean value of $a_n(k)$ as $n$ varies. For $k\ge1$, write

$$M_n(a_n(k))=\frac{6e_k}{\pi^2}.$$

Möller's conjecture.

$$0\le e_k\le\frac12,$$

and

$$(-1)^k(e_k-e_{k+1})>0.$$

The source attributes these two proposed properties to Möller and later asks specifically whether the first is true; it gives no resolution status for the combined assertion.