Fractional-sum counting conjecture for singular-series analysis

Let $\delta>0$ and $Q>1/\delta$. For fixed $k$, let $S$ be the number of tuples

$$(q_1,\ldots,q_k,a_1,\ldots,a_k)$$

with $q_i\in[Q,2Q]$, $a_i$ taken modulo $q_i$, satisfying $\left\|a_i/q_i\right\|\leq\delta$ for every $i$ and $\sum_i a_i/q_i\in\mathbb Z$. Fractional-sum counting conjecture. For every $\varepsilon>0$,

$$S\ll\begin{cases}Q^{k+\varepsilon}\delta^{k/2}&k\text{ even},\Q^{k+\varepsilon}\delta^{(k+1)/2}&k\text{ odd}. \end{cases}$$

This estimate is proposed as a key counting input for controlling the sums arising from singular-series moments; the supplied text gives no evidence that it has been resolved.