Self-avoiding loop measure scaling-limit conjecture

Let $L$ be a regular periodic planar lattice, such as the square, triangular, or hexagonal lattice, and let $bb>1$ be the lattice-dependent connective constant, so that the number of self-avoiding paths of length $n$ starting at the origin behaves like $bb^{n+o(1)}$ as $n\to\infty$. For mesh size $\delta>0$, let $\mu^{\delta}$ be the discrete measure on self-avoiding loops drawn on $\delta L$ that assigns weight $n^{-\lambda}$ to each loop with $n$ steps.

Self-avoiding loop scaling-limit conjecture.

$$\mu^{\delta}\text{ converges as }\delta\to0\text{ to a limiting measure on self-avoiding loops}.$$

This is presented as a motivation for understanding long self-avoiding walks and related conjectures. The source gives no resolution status.