Covariance perturbation invariance conjecture for GMC regularizations

Let $\widetilde{\omega}_{\mu,\varepsilon}(s)$ be a Gaussian process whose covariance agrees with the covariance in Eq. up to $O(\varepsilon)$ as $\varepsilon\to0$, and whose mean agrees with Eq.. Covariance perturbation invariance conjecture. Its exponential functional gives rise to the same GMC measure as the construction in Section 2. The claim is motivated by mesoscopic statistics of Riemann zeros, but no proof or resolution is supplied here.