The problem of characterizing $1$-safe Petri nets generating all the $2^n$ binary $n$-vectors as marking vectors exactly once is an open problem \cite{ksa2}. In this note, we completely settle a part of this problem, viz., to determine minimal such Petri nets, `minimal' in the sense that the depth of their reachability tree is minimum possible. In fact, we show here that such a $1$-safe Petri net has a unique structure.