Smooth-number approximation conjecture for Dirichlet polynomials

Let $P(n)$ denote the largest prime factor of $n$, and let $\Psi(x,y)$ denote the number of integers smaller than $x$ with $P(n)\leq y$. For a constant $A>0$, set

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

Smooth-number approximation conjecture. There exists a constant $A>0$ such that, for any $1\le x\le T$ and $2T\le t\le 5T$, uniformly as $T\to\infty$,

$$\sum_{n\le x}\frac{1}{n^{it}}=\sum_{\substack{n\le x\P(n)\le y}}\frac{1}{n^{it}}+o\left(\Psi(x,y)\right).$$

This conjecture asserts that the Dirichlet polynomial is asymptotically determined by its $y$-smooth terms in the stated range. It is described as similar to a conjecture of Granville and Soundararajan on character sums and would imply the stated asymptotic for the maximum of $\left|\zeta^{(\ell)}(1+it)\right|$.