The freezing hypothesis for the exponential functional of the GFF

Let $X$ be the normalized exponential functional introduced in the source, and let $q$ be in the range where the displayed moments exist. Freezing hypothesis. For $\beta>1$,

$$\mathbf{E}\left[X^{q/\beta}\right]\Gamma\left(1-\frac q\beta\right)=\left.\mathbf{E}\left[X^{q/\beta}\right]\Gamma\left(1-\frac q\beta\right)\right|_{\beta=1}.$$

This is the freezing assumption that extends the high-temperature expression beyond the critical point and underlies the proposed maximum-distribution asymptotics. It was formulated in earlier work on logarithmically correlated fields; the supplied text does not establish it.