Brownian motion produced by the Dambis-Dubins-Schwarz theorem need not be independent of its clock

The Brownian motion $B$ in the Dambis-Dubins-Schwarz theorem representation $M_t=B_{[M{]}_t}$ need not be independent of the bracket clock. Let $W$ be standard Brownian motion and takeThe Gaussian fourth moment and the Brownian covariance kernel give $\mathbb{E}[(W_t^2-t)W_s^2]=2s^2$ for $s\le t$, and henceIf the entire driving Brownian motion $B$ were independent of the clock $A$, conditioning on $A_t$ would give $\mathbb{E}[B_{A_t}{A}_t]=0$, contradicting this identity. Gaussian increments in bracket time therefore do not justify an independent random-clock model.