The persistence asymptotic conjecture for fractional Brownian motion

Let $B^H$ be fractional Brownian motion with Hurst parameter $H\in(0,1)$, let $A_t^H=\int_0^tB_s^H\,\mathrm{d}s$, and define $T_1^H=\inf\{t\ge0:A_t^H>1\}$. The fractional Brownian persistence conjecture. For every $H\in(0,1)$,

$${\mathbb{P}}[T_1^H>t]\asymp t^{H-1}.$$

Molchan proved the corresponding logarithmic exponent, while the two-sided estimate with comparable constants is presented here as open.