Bounds for Dirichlet $L$-functions on the critical line

Question

I am interested in bounds on the constants $A,B$ such that $$L\big(\tfrac{1}{2}+it,\chi\big)\ll_\varepsilon q^{A+\varepsilon}(|t|+1)^{B+\varepsilon},$$ and was curious if any developments have been made since the work of Huxley and Watt in 2000. In particular, I am interested in fixed-order Dirichlet characters $\chi\bmod{q}$, though any bound applying to general Dirichlet characters would probably suffice.