Conjecture on smooth approximations to oscillatory sums

For $x\geqslant1$ and $y\geqslant1$, write $\Psi(x,y;t)=\sum_{\substack{n\leqslant x\P^{+}(n)\leqslant y}}n^{-it}$.

Smooth approximation conjecture. There exists a constant $A>0$ such that, uniformly for $1\leqslant x\leqslant T$ and $2T\leqslant t\leqslant5T$,

$$\sum_{n\leqslant x}\frac{1}{n^{it}}=\Psi(x,y;t)+o\bigl(\Psi(x,y)\bigr),\qquad\text{as }T\to\infty,$$

where $y=(\log T+\log^2x)(\log_2T)^A$.

The source presents this as the analogue of the Granville–Soundararajan conjecture and notes that it could imply sharpness of the constant in an extreme-value result for $\zeta(1+it)$. No resolution is supplied.