Poisson-Dirichlet statistics for the log-modulus of the Riemann zeta function on expanding intervals near the critical line

For $T > 0$ large, let $τ$ be uniformly distributed on $[T,2T]$ and let $σ= σ(T)$ be a deterministic real part approaching $1/2$ from the right. For the Riemann zeta function $ζ$, we study the Gibbs measure with density proportional to $|ζ(σ+ \mathrm{i}τ+ \mathrm{i} h)|^β$ on $I_θ := [ - (\log{T})^θ