Rules as written suggest there is no reason to buy specializations for career skills.
The character can purchase fortune dies for one of the primary characteristics listed on his career card. This is a permanent addition to that characteristic's dice pool and should be noted in the white square next to the characteristic. Fortune dice are cumulative. If a character already has a fortune die in a characteristic from a previous advance, selecting this advance again provides a second die to the characteristic pool.
E.g. Rank 1 character has a career skill (that is based on their career's primary attributes) they want to improve, but they've already max'ed out on expertise die.
Q1: Why would they spend 1 advance on a specialization when they could spend it on an additional fortune die for their primary characteristic?
Q2: It would getting a fortune die for the ability is much more valuable. Is there a scenario where spending the advance on specialization has a greater mechanical benefit than adding a fortune die to the primary characteristic?
Q3: Is there a limit on the number of fortune dice that can be added to a characteristic?
Q4: Is there a limit on the number of specialization dice that can be added to a skill?
Q5: If it's true that specializations on career skills provide materially less benefit than fortune dice added to characteristics, under what circumstances do people find themselves using them (other than at character creation and career completion)?