The supremum of the Jensen-Shannon chain-rule ratio R = IJS (X;Y, Z) / IJS (X;Y) + IJS (X;Z|Y) is achieved on binary alphabets: C* = sup R = 1. 4903. . . with |X|=|Y|=|Z|=2. Proof via Dinkelbach linearization and double concave envelope reduction to two-point martingales. The sharp constant satisfies an explicit KKT system and is approached but not attained — only in a nested rare-event limit ε→0.
Alex B. Shvets (Sun,) studied this question.