Infinitude conjecture for row-dominant characters

Let $p$ be a prime and let $\chi$ be a character modulo $p$. Say that $\chi$ is row-dominant when it satisfies the row-dominance condition defined in the paper.

Infinitude conjecture. There are infinitely many row-dominant characters.

The conjecture is motivated by a heuristic that the values $\phi_\chi(p)$ have substantial variance, together with numerical evidence. The source does not provide a proof or a resolution.