We consider the optimal stopping problem $v^{(\varepsilon)}:=\sup_{\tau\in{\cal T}_{0,T}}{\bf E}\,B_{(\tau-\varepsilon)^+}$ posed by Shiryaev at the International Conference on Advanced Stochastic Optimization Problems organized by the Steklov Institute of Mathematics in September 2012. Here $T>0$ is a fixed time horizon, $(B_t)_{0\le t\le T}$ is the Brownian motion, $\varepsilon\in[0,T]$ is a constant, and ${\cal T}_{\varepsilon,T}$ is the set of stopping times taking values in $[\varepsilon,T]$. The solution of this problem is characterized by a path dependent reflected backward stochastic differential equation, from which the continuity of $\varepsilon \to v^{(\varepsilon)}$ follows. For large enough $\varepsilon$, we obtain an explicit expression for $v^{(\varepsilon)}$, and for small $\varepsilon$ we have lower and upper bounds. The main result of the paper is the asymptotics of $v^{(\varepsilon)}$ as $\varepsilon\searrow 0$. As a by-product, we also obtain Levy's modulus of continuity result in the ...