I'd say definitely connect ground on the strings together as well. They'll be reading data from the voltage between GND and Data wires - so if GND can be at different voltage in each string (which is possible because of the current draw) then they may not work correctly.
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I'd say definitely connect ground on the strings together as well. They'll be reading data from the voltage between GND and Data wires - so if GND can be at different voltage in each string (which is possible because of the current draw) then they may not work correctly.