Supposedly, a Black-Scholes riskless hedge will break down if the volatility is non-constant. However, a random walk with any sigma could produce any price history with some non-zero probability. If there exists a price history that will break your riskless hedge, then you never had one to begin with, because a random walk with a constant sigma could have produced that price history and broken your hedge.
You might think you can look at a price history and conclude that the volatility changed on a certain day, but in reality there is always a chance that a random walk with constant sigma produced that price history. If a hedge is riskless given constant volatility, then it can't fail under any price history, even price histories that will cause investors to say that the volatility has changed.
What is the resolution to this apparent contradiction?