2-variation Of Continuous Price Processes

This question beclongs to continuous-time game-theoretic probability and is asked in [Vovk 2007], Section 5.

It is known that, almost surely, the strong $p$-variation $\text{var}_p$ of non-constant continuous price processes is finite when $p>2$ and infinite when $p<2$. What is the situation when $p=2$?

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