In this manuscript, we address open questions raised by Dieker \& Yakir (2014), who proposed a novel method of estimation of (discrete) Pickands constants $\mathcal{H}^δ_α$ using a family of estimators $ξ^δ_α(T), T>0$, where $α\in(0,2]$ is the Hurst parameter, and $δ\geq0$ is the step-size of the regular discretization grid. We derive an upper bound for the discretization error $\mathcal{H}_α^0 - \mathcal{H}_α^δ$, whose rate of convergence agrees with Conjecture 1 of Dieker & Yakir (2014) in case $α\in(0,1]$ and agrees up to logarithmic terms for $α\in(1,2)$. Moreover, we show that all moments of $ξ_α^δ(T)$ are uniformly bounded and the bias of the estimator decays no slower than $\exp\{-\mathcal CT^α\}$, as $T$ becomes large.