Classify the first-layer beta-near-critical CLE3 law

Determine whether, for plus boundary conditions in a finite domain, the beta-near-critical scaling limit of the first layer of CLE3 is singular or absolutely continuous with respect to critical CLE3^1.

Background

The paper distinguishes the full nested CLE3 from its first layer, which consists of the outermost spin interfaces. For the beta-near-critical perturbation of the inverse temperature, the authors do not know whether the law of this first-layer limit changes in a singular way or merely by an absolutely continuous change of measure relative to critical CLE31.

This question is explicitly presented as unresolved even in the plus-boundary-condition case.

References

In the case of $+$ boundary conditions, is the near-critical limit for the first layer of the $CLE_3$, i.e. $\Gamma1$ in the notations of this text, singular or absolutely continuous w.r.t. a critical $CLE_31$? (We do not even know what should be expected here).

— The critical Ising magnetization field can be reconstructed from its $+/-$ interfaces  (2610.02064 - Cahen et al., 1 Oct 2026) in Section "Near-critical limit(s) for nested--CLE3," Question following Open Problem 2, item 1