Technical conjecture on truncated divisor sums near rational approximants

Let $\mathcal{K},\lambda,\zeta,c>0$ be constants with $\lambda<1$. Let $N\text{?}1$ and suppose that $p/q\text{?}\mathbb{Q}$, $(p,q)=1$, is of type $(2+\lambda,\mathcal{K})$. Assume

$$N^{\frac{3}{2+\lambda}}\le q\le N^{\frac{3(1+\lambda)}{2+\lambda}}.$$

Technical divisor-sum conjecture. Then

$$\sum_{|r|\le \frac{\zeta q}{N}}\sum_{pm\equiv r(q)}\tau_{cN,N}(m)\sim 2\zeta cN$$

as $N\to\infty$, uniformly in $q$ and $p$.

The conjecture is intended as a technical input for decomposing the arithmetic expression governing quadratic pair correlation. The source provides no proof or resolution status.