General oscillation conjecture for weighted multiple sine sums

Let $k\geq 1$, and let $a_1,\ldots,a_k$ and $p_1,\ldots,p_k$ be the parameters used in the paper. Define

$$\mathbb{S}(a_1,a_2,\dots,a_k;p_1,p_2,\dots,p_k;x):={\sum_{n_1n_2\cdots n_k\leq x}}^{\prime} n_1n_2\cdots n_k\prod_{j=1}^k\sin\left(\frac{2\pi n_j a_j}{p_j}\right).$$

General weighted sine-sum oscillation conjecture.

$$\varlimsup_{x\to\infty}\frac{\mathbb{S}(a_1,a_2,\dots,a_k;p_1,p_2,\dots,p_k;x)}{x^{(3k-1)/(2k)}}=+\infty,$$ $$\varliminf_{x\to\infty}\frac{\mathbb{S}(a_1,a_2,\dots,a_k;p_1,p_2,\dots,p_k;x)}{x^{(3k-1)/(2k)}}=-\infty.$$

The conjecture generalizes the two-factor weighted sine-sine assertion. It is motivated by the proved $\Omega$-theorem for the associated multiple character-divisor sums, while the required oscillation for their trigonometric linear combination remains unproved.