AC has effectively divided the concierge service into three tiers in International business class:
(a) Full Service (Phone, e-mail, in person) --> AC SE
(b) Limited Service based on availability --> Those non-AC SE flying on paid International business class tickets. Eligibility on the domestic connector restricted to Signature Class flights only. Access to services at the concierge office at select airports not allowed.
(c) No Service --> I class, eUpgrade, LMU.
That is only two classes, unless you wish to enumerate all the other ways one doesn't get concierge access.