Mixed Granville–Soundararajan conjecture for twisted character sums

Let $\chi$ be a non-principal character modulo $q$, let $1\leqslant x\leqslant\min\{q,T\}$, and let $2T\leqslant t\leqslant5T$. Write $\chi_0$ for the principal character modulo $q$, and let $P^{+}(n)$ denote the largest prime factor of $n$.

Mixed conjecture. There exists a constant $A>0$ such that, uniformly in these ranges,

$$\sum_{n\leqslant x}\frac{\chi(n)}{n^{it}}=\sum_{\substack{n\leqslant x\P^{+}(n)\leqslant y}}\frac{\chi(n)}{n^{it}}+o\bigl(\Psi(x,y;\chi_0)\bigr),\qquad\text{as }q\to\infty,\ T\to\infty,$$

where $y=(\log(qT)+\log^2x)(\log_2(qT))^A$.

This combines the character-sum and oscillatory smooth-approximation conjectures. The source gives no evidence of a proof or disproof.