Proof of the conjecture about inequality of $1/H$-variations of fractional Brownian motion along the equidistant and the Lebesgue partitions

We give an analytic proof of the comparison conjectured by Das, Łochowski, Matsuda and Perkowski for the fractional Brownian level-crossing constant: for standard fractional Brownian motion, $c_H>\mathbb{E}|Z|^{1/H}$ when $0<H<1/2$ and $c_H<\mathbb{E}|Z|^{1/H}$ when $1/2<H<1$, where $Z\sim N(0,1)$.