Sharp maximal bound for continuous martingales

Let XX, YY be continuous-path martingales satisfying the condition [X,X]t≥[Y,Y]t[X,X]t≥[Y,Y]t for all t≥0t≥0. We prove that ‖supt≥0Yt‖1≤32‖supt≥0|Xt|‖1 and the constant 3/23/2 is the best possible.