Suppose X_1,X_2,...,X_n are n Brownian motions with respect to the same filtration such that X_1 is independent of X_j for all j=2,...,n. Is it true that X_1 is independent of (X_2,...,X_n)?
This is clearly false. Take X independent of Y and Z=integral of sgn(X_s)dY_s. Then if your result is true we would have X independent of the cross vriation of Y and Z which is integral of sgn(X_s)ds. This is false.