Conjecture on the oscillation of the cosine-sine divisor sum

Let

$$\mathbb{C}\mathbb{S}(x):={\sum_{nm\leq x}}^{\prime}\cos\left(\frac{2\pi na}{p}\right)\sin\left(\frac{2\pi mb}{q}\right),$$

with the parameters and primed summation convention used in the paper. Cosine-sine oscillation conjecture.

$$\varlimsup_{x\to\infty}\frac{\mathbb{C}\mathbb{S}(x)}{x^{1/4}}=+\infty, \qquad \varliminf_{x\to\infty}\frac{\mathbb{C}\mathbb{S}(x)}{x^{1/4}}=-\infty.$$

This predicts unbounded oscillation at scale $x^{1/4}$. The motivation is that available $\Omega$-theorems control the constituent character-divisor sums, but do not directly yield an $\Omega$-theorem for their linear combination.