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.
Yep, just read that in the rules. Love that they wait until so long in (after casually referencing first turn eligibility) to say, “oh by the way he gets a Prior Consul marker”.