The prime-model conjecture for the maximum of the truncated zeta-function

Let $Z_X(s)$ denote the truncated zeta-function used in the paper, and let $A>0$ be fixed. Truncated-zeta maximum conjecture. If $2<X<\log^A T$, then, as $T\to\infty$,

$$\max_{t\in[0,T]}|Z_X(\tfrac12+{\mathrm{i}}t)|=\exp\left((1+o(1))\sqrt{\tfrac12\log T\log\log T}\right).$$

The conjecture comes from modelling $Z_X$ by characteristic polynomials of random unitary matrices and is intended to support the corresponding maximum-size conjecture for $\zeta(\tfrac12+{\mathrm{i}}t)$.