Water flows steadily through a pipe. At point $A$, the pipe has radius $r$ and is located at an elevation $z_A = 0$. At point $B$, the pipe is at an elevation $z_B = h$ and has narrowed to a radius $r/2$. Vertical open-topped glass tubes are connected to the pipe at points $A$ and $B$, acting as piezometers, and water rises in both tubes. The velocity of the water at point $A$ is $v$. Let $H_A$ and $H_B$ be the absolute elevations of the water surfaces in the two tubes, measured from the $z=0$ reference level. Assuming the water acts as an ideal fluid and the pressure is high enough that water rises in both tubes, what is the difference in elevation $H_A - H_B$?