Nobody has mentioned the physical mechanism that draws the water up. Toward (the edge of) a capacitor (or more obviously, to a single electrostatically charged rod), the water molecules (neutral free dipoles) align with the varying field, and (since the field strength now differs between the the two 'poles of the molecule) the molecules have a net attraction in the direction of stronger field.
This being understood, we can reduce the problem by replacing that mechanism with something else (conceptually simpler or just more familiar) that has an exactly equivalent effect. For example, we can replace the charged capacitor with a very dense mass, which also causes the water to rise higher on that side of the U (but by the mechanism of gravity). With no remaining "herrings", it should be obvious that the water would not flow through a connection between the U-arms.. indeed, if it would, then it have already done so through the bottom of the U-tube. Friction is also clearly irrelevent. If the water is forcefully attracted to between the plates, why wouldn't that force also oppose it from flowing elsewhere?
Another way of looking at things would be to imagine an O-shaped tube full of water. One probably wouldn't expect a capacitor around one section to cause the water to start circulating? Though with some electric current leakage..