$\sup B_{1 + t} - B_1$ is independent of $B_1$ for $B$ a brownian motion

Question

Is $\sup_{t \ge 0} B_{1 + t} - B_1$ is independent of $B_1$ for $B$ being a brownian motion? For all $t$ $B_{1+t} - B_1$ is independent of $B_1$ but this isn't enough.

I also can't use that $\sup_{t \ge 0} B_{1 + t} - B_1$ is $\infty$ a.s., because we are using the independence to prove this result.