Determine whether the uniform CLE4 exploration is measurable from the CLE4 loop ensemble

Determine whether the uniform exploration of a CLE4 is almost surely determined by the CLE4 loop ensemble alone.

Background

Werner and Wu introduced the uniform exploration of CLE4, which assigns discovery times to loops and describes the growth of metric balls from the domain boundary. The paper constructs subsequential limits of renormalized CLEκ graph metrics as κ decreases to 4 and proves that the limiting distances from the boundary have the law of the uniform exploration times.

The measurability question asks whether these exploration labels are intrinsic measurable functions of the unlabelled CLE4 loop ensemble, rather than requiring additional randomness or an auxiliary coupling. The paper states that this issue is not resolved within the present article and will follow from its subsequent papers.

References

It was also conjectured in that the uniform exploration is a.s. determined by the $\CLE_4$. That this holds will be a consequence of the sequels to the present paper.

— The conformally invariant metric on CLE$_4$ I: subsequential limits of the non-simple CLE graph metric  (2609.04138 - Kammerer et al., 3 Sep 2026) in Section 1, Subsection 1.1 (Overview)