Conjecture on the convergence rate of discretized Pickands constants

Let $\mathcal H_\alpha(T)$ be the finite-horizon Pickands constant, let $\mathcal H^\eta_\alpha(T)$ be its discretized approximation with mesh size $\eta$, and let $\mathcal H_\alpha$ and $\mathcal H^\eta_\alpha$ denote their corresponding infinite-horizon quantities. For fixed $T>0$,

Convergence-rate conjecture.

$$\lim_{\eta\downarrow0} \eta^{-\alpha/2}[\mathcal{H}_\alpha(T) - \mathcal{H}^\eta_\alpha(T)] \in(0,\infty).$$

We also have

$$\lim_{\eta\downarrow0} \eta^{-\alpha/2} [\mathcal{H}_\alpha - \mathcal{H}^\eta_\alpha] \in(0,\infty).$$

The conjecture gives the rate at which the discretized constants converge to the continuous ones and is explicitly stated to be outside the scope of the paper's proof; it motivates the regression-based numerical approach.