Macroscopic-loop support conjecture for the zero-curvature loop measure

Let ${\mathbb P}_{LC^0}$ be the limiting loop measure associated with the zero-curvature scaling, and let $\Omega_1$ denote the event that the loop has nonzero area. Macroscopic-loop support conjecture. The measure ${\mathbb P}_{LC^0}$ is supported on $\Omega_1$, that is

$${\mathbb P}_{LC^0}\left(\text{the area of the loop is zero}\right)=0\,.$$

This conjecture asserts that the zero-curvature scaling limit is still supported on macroscopic loops, despite arising as a metric rescaling limit. The source notes that the corresponding statement for ${\mathbb P}_{LC}$ is known, while the assertion for ${\mathbb P}_{LC^0}$ is not established.