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 () that the Peano curve converges to a space-filling SLE loop, where $y=1-2\cos (4\pi /\kappa )$, corresponding to . We argue that the same should hold for , which corresponds to .