All L_∞-spaces are separably injective, kind of

How reasonable is the conjecture that _∞ /C[0,1] ℓ ∞ / C [ 0 , 1 ] separably injective? Formulated in homological terms this amounts to asking whether Ext²(S, C[0,1])=0 Ext 2 ( S , C [ 0 , 1 ] ) = 0 for every separable space. We wheel around this question and obtain new connections between Ext² Ext 2 -results and classical Banach space theorems of Johnson–Zippin, Kalton and Lindenstrauss–Pełczyński.