Distribution of the cotangent of the SLE tip argument

Let $\kappa<8$, and consider the backward $SLE_{\kappa}$ flow and its $SLE_{\kappa}$ trace. At any fixed deterministic time $t>0$, let the tip of the trace be viewed through the cotangent of its argument, and write $x$ for this real-valued random variable. The cotangent-tip distribution conjecture. The law of $x$ has density

$$\mu(dx)=\frac{C\,dx}{(1+x^2)^{4/\kappa}},$$

where $C$ is a normalizing constant. The paper proves the same closed-form stationary distribution along a suitable sequence of random times, but identifying it with the distribution at every fixed deterministic time remains conjectural.