Things work out much more nicely if you build the latches underneath the register rather than on top, as the paths that are split going through the latches are shorter.
Link: jstumble
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Nice. And here, at the cost of one extra part, is a slight revision of your version that terminates after no more than eight balls:
https://www.lodev.org/jstumble/?board=0e0reellleeixx0eerxxleerllfxrfgalgaeearaleeirleel_16_16
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