 Terminal correlation does not identify adapted stock volatility

The stocks $S_t=e^{B_t-t/2}$ and $S_t^{\prime }=(1+S_t)/2$ start at one, are true square-integrable zero-rate martingales, and have perfect positive correlation at each positive time. Their bounded multiplicative volatilities are $1$ and $S_t/(1+S_t)$, which differ throughout. The additive intercept is what allows the same correlation but different stochastic-exponential coefficients.