Centered GMC measure moment-limit conjecture

Let $\omega_{\mu,\varepsilon}(u)$ be the regularized Gaussian field and define its centered version by

$$\widetilde{\omega}_{\mu,\varepsilon}(u)\triangleq\omega_{\mu,\varepsilon}(u)-\omega_{\mu,\varepsilon}(0).$$

For $-1/\mu-1/2<\operatorname{Re}(q)<2/\mu$, and for a test function $\varphi$, Centered GMC measure conjecture.

$$\lim_{\varepsilon\to0}e^{\mu(\log r(\varepsilon)-1)q(q+1)/2}{\bf E}\left[\left(\int_0^1\varphi(u)e^{\widetilde{\omega}_{\mu,\varepsilon}(u)}\,du\right)^q\right]={\bf E}\left[\left(\int_0^1r(u)^{\mu q}\varphi(u)\,M_\mu(du)\right)^q\right].$$

This proposes the limiting moment identity for the centered regularization and the GMC measure; the supplied source marks it as an open question.