Generalized intersection-exponent cascade conjecture for SLE restriction measures

Fix $\kappa\in(0,4]$, let $n\geq 1$, and let $\alpha_1,\ldots,\alpha_n$ denote the parameters of the restriction measures. Define

$$V_{\kappa}(x)=\sqrt{16\kappa x+(4-\kappa)^2}-(4-\kappa),\qquad V_\kappa^{-1}(x)=\frac{x^2+2(4-\kappa)x}{16\kappa}.$$

The generalized intersection exponents are the exponents governing the non-intersection probabilities of independent restriction measures as their marked points tend to each other. Generalized intersection-exponent cascade conjecture. The half-plane intersection exponents are

$$\widetilde\xi_{\kappa}(\alpha_1,\ldots,\alpha_n)=V_\kappa^{-1}\left(\sum_{j=1}^n V_\kappa(\alpha_j)\right),$$

and the whole-plane intersection exponents are

$$\xi_\kappa(\alpha_1,\ldots,\alpha_n)=\eta_\kappa\!\left(\widetilde\xi_\kappa(\alpha_1,\ldots,\alpha_n)\right)=\frac{1}{32\kappa}\left(\sum_{j=1}^n V_\kappa(\alpha_j)\right)^2-\frac{(4-\kappa)^2}{8\kappa}.$$

The paper explicitly states that it does not establish rigorous statements about these exponents; the formulas are conjectural and are based on previously computed exponents and an analogue of the cascade relation for the standard case $\kappa=8/3$.