It does, that is why they are the only ones eligible for Censor turn 1.
3.01.6 TEMPORARY ROME CONSUL: Draw a Mortality Chit from the cup. If the ID# drawn does not match an existing Senator put it to the side and draw again until a matching ID# is drawn. The matching ID# is Temporary Rome Consul until new Consuls can be elected during the first Senate Phase. Place the Rome Consul and Prior Consul markers on his card and add 5 to his Influence. Place all the Mortality Chits back into the cup.