The first examples of the Gewehr 43 did not enter service until late 1943. The DAK fought it's last battle on the 28th-30th of April, 1943. (
Germany's Infantry Weapons 1939-45, page 32.) The MP44 did not enter production until early 1944, hence the model number. The early assault rifle mentioned is most likely the MKb 42, which was used for troop trails during this period. It is very similar in appearance to the MP43/44, neither of which were in operation at this time.