SLE trace minimality conjecture

For $\kappa\in[0,8]$, let $\mathrm{SLE}_\kappa$ denote Schramm–Loewner evolution with parameter $\kappa$, and let its trace be the associated random curve. SLE trace minimality conjecture. Almost surely, the $\mathrm{SLE}_\kappa$ trace is minimal for conformal dimension; equivalently, its conformal dimension is

$$1+\frac{\kappa}{8}.$$

The value $1+\kappa/8$ is the known almost-sure Hausdorff dimension of the trace for $\kappa\in[0,8]$. The conjecture asks whether quasisymmetric deformations can never lower this dimension.