The bounds $\hbar_{p,X}\lesssim(β_{p,X})^2$ and $β_{p,X}\lesssim(\hbar_{p,X})^2$ are sharp

It was proved in the 1980s by Burkholder and Bourgain that, for any Banach space $X$ and $1<p<\infty$, the UMD$_p$ property for $X$ is equivalent to boundedness of the Hilbert transform on $L^p(\R;X)$, and that the UMD constant $β_{p,X}$ and the Hilbert transform constant $\hbar_{p,X}$ are related b