Almost-sure multifractal spectrum conjecture for SLE

Let $\bar\beta(t)$ denote the average spectrum of SLE, and let $t_{\min}$ and $t_{\max}$ be the two points such that a tangent to $\bar\beta(t)$ intersects the $y$-axis at $-1$. The almost-sure SLE spectrum conjecture. The almost-sure spectrum equals $\bar\beta(t)$ for $t_{\min}\leq t\leq t_{\max}$ and continues as the corresponding tangent lines for $t<t_{\min}$ and $t>t_{\max}$.

The paper explains that the average spectrum cannot itself be the almost-sure spectrum because it violates Makarov's characterization of possible spectra. It states that explicit formulas for $t_{\min}$, $t_{\max}$ and the tangent lines are given elsewhere in the paper, while the question about spectra of individual SLE realizations otherwise remains open.