The active spanning-tree Peano-curve conjecture

Let $\Lambda$ be the triangular, hexagonal, or square lattice. Suppose either $y=0$ and $z>0$, or $y\in[0,1]$ and $z\in[1,\infty)$. Let $T$ be a spanning tree on $\Lambda$ sampled according to the law defined by the active-tree model, and let $\lambda$ be its associated Peano curve. Let $\chi=\chi(y,z)$ be the constant from the source's variable-parameter theorem. The active spanning-tree Peano-curve conjecture. The curve $\lambda$ converges in law in the scaling limit to $\operatorname{SLE}_\kappa$, where $\kappa\geq8$ satisfies

$$-\cos\mathopen{}\mathclose\bgroup\originalleft(\frac{4\pi}{\kappa}\aftergroup\egroup\originalright)=\begin{cases}-\dfrac{z}{1+z},&y=0,\\-\dfrac{(z-y)\chi}{(y+1)(z+1)+(z-y)\chi},&(y,z)\in[0,1]\times[1,\infty),\end{cases}$$

This expands the earlier active spanning-tree conjecture to a two-parameter family and to the specified lattices; the scaling-limit assertion remains open.