Active spanning trees and SLE

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 (1Peano curve converges to a space-filling SLEκ loop, where y=1−2cos(4π/κ), corresponding to 4<κ≤8. We argue that the same should hold for 0≤y<1, which corresponds to 8<κ≤12.