The level one conjecture for the divisor function

Let $q\geq 1$ be a composite number, and let $W(x)$ be a smooth test weight supported on $[1,2]$. Its Mellin transform is

$$\widehat{W}(s)=\int_0^\infty W(x)x^{s-1}\,dx.$$

Assume that, for every fixed $A>0$ and every positive integer $\ell$,

$$\widehat{W}(\sigma+it)\ll_\ell \frac{1}{(1+|t|)^\ell}$$

uniformly for $|\sigma|\leq A$. The level one conjecture. For every $\epsilon>0$, there exists $\delta=\delta(\epsilon)>0$ such that

$$\sum_{\substack{n\\ n\equiv a\pmod q}}\tau_2(n)W\left(\frac{n}{X}\right)=\frac{1}{\varphi(q)}\sum_{\substack{n\\ (n,q)=1}}\tau_2(n)W\left(\frac{n}{X}\right)+O_\delta\left(\frac{X^{1-\delta}}{\varphi(q)}\right)$$

uniformly for $(a,q)=1$ and $q\leq X^{\theta_{W,2}-\epsilon}$, where $\theta_{W,2}=1$. This conjectures level of distribution one for the smoothed divisor-function progression sum; the paper establishes the corresponding prime-modulus result only at level $2/3$, while the composite-modulus level-one assertion remains open.