Purgatory and Plenary Indulgences?

In regards to the purification of the soul before entering heaven; suppose a person has a desire that is sinful and has struggled with it for many years to the point that it has become habitual. The person realizes the struggles to overcome the bad habits and prays constantly and goes to frequent communion. At the moment of death, the person receives the sacraments and is repentant of their sins; however, the desires have remained. Do the desires also have to be burned off in purgatory before one can enter heaven; even though the person was absolved of the sin, receives the apostolic pardon and dies in the state of grace? I know that by receiving a plenary indulges the temporal punishment of sin is removed, but what about the desires that have remained?