Heuristic least-nonresidue bound from a quadratic twist

Let $\xi$ be an odd primitive Dirichlet character modulo $k$ with $3\nmid k$, and define

$$\chi(n):=\xi(n)\left(\frac{n}{3}\right).$$

Let $n_\xi$ denote the least positive integer $n$ with $\xi(n)\neq 1$, and let $M(\chi)$ denote the maximum size of the partial sums of $\chi$. Heuristic conjecture. One expects that

$$\log n_\xi\leq\left(\frac{\pi}{e^\gamma}+o(1)\right)\frac{M(\chi)}{\sqrt{k}}.$$

This is presented as a heuristic consequence suggesting that the preceding bounds are not optimal; it would relate the least nonresidue directly to one specific quadratic twist and could sharpen estimates for least nonresidues if proved.