The large-values limsup conjecture for summatory $k$-free character functions

Let $k\geq 2$ be an integer, let $\chi$ be a real non-principal Dirichlet character modulo $q$, and let $f=\mu^{(k)}\chi$ or $f=\mu^{(k)}g_\chi$. Then the large-values limsup conjecture asserts that there exists a constant $C_k>0$ such that

$$\underline{\overline{\lim}}_{x\to\infty}\frac{\sum_{n\leq x}f(n)}{x^{\frac{1}{2k}}(\log\log x)^{\frac{1}{2}-\frac{1}{2k}}(\log\log\log x)^{\frac{1}{4k}}}=\pm C_k.$$

This is presented as a consequence of the stated large-deviation conjectural picture and Ng's analysis; its validity remains open.