Klurman–Tenenbaum–Teräväinen conjecture on modified character partial sums

Let $\chi$ be a primitive Dirichlet character, and let $f$ be a completely multiplicative modification of this character such that $f(p)=\chi(p)$ for all but a finite subset $S$ of primes $p$, where for each prime $p\in S$, $|f(p)|=1$.

Klurman–Tenenbaum–Teräväinen conjecture. The partial sums satisfy

$$\sum_{n\leq x}f(n)=\Omega((\log x)^{|S|}).$$

This conjecture strengthens the known lower-bound results for partial sums of modified Dirichlet characters. The source notes that the preceding formulation without primitivity is false in general, because modifying a non-primitive character can produce a character of smaller modulus; the primitive case remains the conjectural statement here.