Scaling-limit conjecture for the canonical evaporated UST component

Let $({\bf T},r)$ be a rooted uniform spanning tree of a graph, and obtain ${\bf t}^{<n}(r)$ by successively removing outgoing edges at independent uniformly chosen non-root vertices until the component containing $r$ first has size less than $n$. For a tree of diameter smaller than the torus side, let ${\sf Canonical}({\bf t}^{<n}(r))$ denote its canonical embedding in $\mathbb{Z}^2$. The evaporated-tree scaling conjecture. If $N(n)$ is a sequence of integers satisfying $\limsup n/N(n)<1$ and $G={\sf Torus}(N(n))$, then

$$\frac{{\sf Canonical}({\bf t}^{<n}(r))}{\sqrt n}$$

converges in distribution, for the Hausdorff metric on compact subsets of $\mathbb{R}^2$, to a limiting compact set $K$ with Lebesgue measure $1$ that is simply connected. The conjecture predicts that repeated edge evaporation does not lose macroscopic area in the scaling limit; the motivation given is that simulations show most removals discard only very small components. No proof or resolution is supplied.