= Solution
Continuity gives existence of a minimizer. If there were two, choose a rational $q$ strictly between them. Then the minima on $[0,q]$ and $[q,1]$ would coincide. Conditional on $\mathcal F_q$, the latter equals $B_q$ plus the minimum of an independent Brownian motion on $[0,1-q]$, whose distribution is continuous by the <Brownian reflection principle>. Thus equality has conditional probability zero. Taking the countable union over rational $q$ proves almost-sure uniqueness.
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