Khokhlov-Molchan conjecture for integrated fractional Brownian motion

Let $B^H$ be fractional Brownian motion with Hurst parameter $H\in(0,1)$, and define

$$I_t^H=\int_0^tB_s^H\,\mathrm{d}s.$$

There is a function $\rho(H)>0$ such that

$${\mathbb{P}}\left[I_s^H\le1,\ \forall s\in[0,t]\right]=t^{-\rho(H)+o(1)}.$$

Khokhlov-Molchan conjecture. One has

$$\rho(H)=H(1-H).$$

The value is motivated by numerical simulations and is not proved in the paper.