Immune to Interruption protects from attacks (mysteries, chases, standard) that cause interruption in addition to their damage. Debuffs that cause mysteries to be unable to execute (immobile, sleep, etc), will still cause the mystery to be "cancelled", which in effect interrupts it. Even if that debuff is reflected.

Anxiety is defined on Indra, "Units under ANXIETY will take 10% of their current HP whenever they execute any actions. ANXIETY cannot be dispelled and neglects immunity."