The conjectured inequality for the fractional Brownian variation constant

Let $B^H=(B_t^H)_{t\geq 0}$ be fractional Brownian motion with Hurst parameter $H\in(0,1)$, and let $\mathfrak{c}_H$ denote the constant governing its $(1/H)$-th variation along deterministic partitions. The quantity $\mathbb{E}[|B_1^H|^{1/H}]$ is the corresponding constant for uniform Lebesgue partitions.

Variation-constant conjecture. For $H\neq 1/2$, the $(1/H)$-th variation along deterministic partitions differs from that along uniform Lebesgue partitions, more precisely,

$$\begin{cases} \mathfrak{c}_H > \mathbb{E}[|B_1^H|^{1/H}] & \text{if } H<1/2,\\ \mathfrak{c}_H < \mathbb{E}[|B_1^H|^{1/H}] & \text{if } H>1/2. \end{cases}$$

If true, this would show that $\mathfrak{c}_H$ captures a non-Markovian feature of fractional Brownian motion. The equality at $H=1/2$ follows from the strong Markov property of Brownian motion, whereas the proposed inequalities for $H\neq 1/2$ remain open.