Active spanning trees and Schramm-Loewner evolution

We consider the Peano curve separating a spanning tree from its dual spanning tree on an embedded planar graph, where the tree and dual tree are weighted by y to the number of active edges, and "active" is in the sense of the Tutte polynomial. When the graph is a portion of the square grid approximating a simply connected domain, it is known (y =1 and y =1 +√{2 } ) or believed (1 <y <3 ) that the Peano curve converges to a space-filling S_{L E κ} loop, where y =1 -2 cos(4 π /κ ) , corresponding to 4 <κ ≤8 . We argue that the same should hold for 0 ≤y <1 , which corresponds to 8 <κ ≤12 .