Extremal bounds for Dirichlet polynomials with random multiplicative coefficients

For $X(n)$ a Steinhaus random multiplicative function, we study the maximal size of the random Dirichlet polynomial $$D_N(t) = \frac1{\sqrt{N}} \sum_{n \leq N} X(n) n^{it},$$ with $t$ in various ranges. In particular, for fixed $C>0$ and any small $\varepsilon>0$ we show that, with high probability, $$\exp( (\log N)^{1/2-\varepsilon} ) \ll \sup_{|t| \leq N^C} |D_N(t)| \ll \exp( (\log N)^{1/2+\varepsilon}).$$