What is the precise relation between level compressibility and the spectral form factor?

In the study of disordered conductors, the level number variance is defined as

$$Sigma_2 (langle n rangle) equiv langle n^2 rangle-langle n rangle^2 ~, $$

where angular brackets denote disorder average. That is, $Sigma_2 $ is the variance of the number of energy eigenvalues in an energy interval that on average contains $langle n rangle$ eigenvalues. For an energy interval of length $L$, the level compressibility $chi$ is then defined as

$$ chi = lim_{langle n rangle to infty} lim_{L to infty} frac{dSigma_2(langle n rangle )}{d langle n rangle }~. $$

In this paper, it is established that the level compressibility can also be expressed as a limit of the spectral form factor, which, for an $N$ by $N$ hamiltonian matrix, is expressed as $K(t)= frac{1}{N} langle lvert sum_{j=1}^N exp(2pi i E_j t) rvert^2rangle$ (see also this paper). In particular, the level compressibility is given by the following limit,

$$ chi = lim_{t to 0}K(t) ~, $$

where $E_j$ are the energy eigenvalues after unfolding. However, from the definition from the form factor it is clear that we formally always have $lim_{t to 0}K(t) = N$, so this cannot be the correct expression for $chi$.

My question therefore is: what is the precise relation between the spectral form factor and the level compressibility? Should we consider $K(t)$ at Thouless time (where $K(t)$ attains a minimum) instead of taking $tto 0$?