We prove there is no polynomial deterministic space simulation for two-way random-tape probabilistic space (Pr$_2$SPACE) (as defined in \cite{BCP83}) for all functions $f : \: \NN \rightarrow \NN$ and all $\alpha \in \NN, \, \prtwospace(f(n)) \not\subseteq \dspace(f(n)^{\alpha})$. This is answer to the problem formulated in op cit., whether the deterministic squared-space simulation (for recognizers and transducers) generalizes to the two-way random-tape machine model. We prove, in fact, a stronger result saying that even space-bounded Las~Vegas two-way random-tape algorithms (yielding always the correct answer and terminating with probability 1) are exponentially more efficient than the deterministic ones.