Let $\chi$ be a Dirichlet character mod $q$ and $\Lambda(n)$ be the von Mangoldt function. Let $c(\chi)=1$ if $\chi$ is the principal character, and zero otherwise. Let $\Theta_\chi$ be the supremum of the zeros of the associated $L$-function $L(s, \chi)$. Define the generalised Chebyshev psi function $\psi(x, \chi):=\sum_{n\leq x} \Lambda(n)\chi(n)$. Do there exist infinitely many $x$ such that $$\psi(x, \chi) - c(\chi)x = \Omega(x^{\Theta_{\chi}-\varepsilon})$$ for each $\chi$ and every $\epsilon>0$ ?