Why does string not wick down?

The wicking is not occuring because of a siphon action. Rather, capillary action is responsible. The abrupt stop at the edge of the cup has two likely explanations; one -- all your solvent evaporated before capillary action wicked solute that far, or two -- The cup preserves a higher relative humidity within, which drops abruptly outside the cup, thus preventing fluid from wicking any significant distance once it gets outside the cup.

Perhaps because oils from your fingers have interfered with the wicking, and that is where you held the string.