A question was recently asked about a generalization of the Fejer-Jackson inequality $$\sum_{k=1}^n \frac{\sin kx}{k}\gt 0 \quad \forall\: n\in\mathbb{Z}^+\: \text{and}\: 0\lt x\lt\pi$$

to denominators of the form $k^\alpha$, which a commenter noted is covered by Theorem A of this paper.

I wonder, are there other references that can address the apparent positivity of the similar sum where the denominator is replaced by the sum of divisors of $k$, $\sigma(k)$?

$$\text{Is}\: \sum_{k=1}^n\frac{\sin kx}{\sigma(k)}\gt 0 \quad \forall\: n\in\mathbb{Z}^+\: \text{and}\: 0\lt x\lt\pi\:?$$

Here is what the sum looks like at large $n$ as a function of $x$:

Can anything else be said about upper or lower bounds in the large $n$ limit?

Thanks.