The other state obligation usually continues until it is satisfied, and Nebraska will see it when you apply for a license here. You generally need a Nebraska policy that can also keep filing to the original state.
More detail
Drivers moving into Omaha or Lincoln from Iowa, Kansas, Colorado, or further are a routine case, and the routine mistake is assuming the move resolved it. It does not. The original state set the obligation, and only that state can close it. What changes is the policy: the vehicle is now garaged in Nebraska, so the policy is rated in Nebraska, and the carrier has to be one that will write here and still transmit certificates to the other state. Not every carrier does both. Before you move, ask the producer that specific question, and keep documentation of the existing filing, because reconstructing the start date of a proof period from another state is slow.