Heath-Brown's upper-bound conjecture for truncated divisor sums

Let $\tau_N^*(m)=\tau_{N,N}(m)$ be the truncated divisor-counting function, and let $u,\rho$ and $q$ be integers with $( u,q)=1$. Fix $\text{?} u\text{?}(0,1)$.

Heath-Brown's upper-bound conjecture. Uniformly for $( u,q)=1$ and $q\text{?}N^{2-\text{?}}$,

$$\sum_{m\equiv \nu(q)}\tau_N^*(m)\ll\frac{\varphi(q)N^2}{q^2},$$

where the implied constant depends only on $\text{?}$.

This is presented as an analogue for $\tau_N^*$ of the Linnik–Vinogradov upper bound for the ordinary divisor function in arithmetic progressions. The source gives no resolution status.