Sharp upper-bound conjecture for alternating products in the modular group

Let $c,q\in\mathbb{Z}_+$ with $q$ even. For $a_1,a_2\in(\mathbb{Z}/c\mathbb{Z})^\times$ and $1\leq H_1\leq H_2\ll c$, consider the integer solutions $(h_1,\ldots,h_q)\in\mathbb{Z}^q$ to the alternating-product equation

$$.$$

Sharpness conjecture. The number of such solutions is at most

$$\ll_q c^{o(1)}\left(H_2^{(q-2)/2}+\frac{(H_1H_2)^{q/2}}{c^3}\right).$$

The preceding lemma proves the same bound without the factor $c^{o(1)}$ as a lower bound, so this conjecture asserts that the displayed lower bound is essentially sharp. It would give the expected order of magnitude for the solution count underlying the paper's bilinear-form estimates with Kloosterman sums.