Motivated by a question of Vincent Lafforgue, the author studies the Banach spaces \(X\) satisfying the following property: there is a function \(\varepsilon\to \Delta_X(\varepsilon)\) tending to zero with \(\varepsilon>0\) such that every operator \(T\colon L_2\to L_2\) with \(\|T\|\le \varepsilon\) that is simultaneously contractive (i.e., of norm \(\le 1\)) on \(L_1\) and on \(L_\infty\) must be of norm \(\le \Delta_X(\varepsilon)\) on \(L_2(X)\). The author shows that \(\Delta_X(\varepsilon) \in O(\varepsilon^\alpha)\) for some \(\alpha>0\) iff \(X\) is isomorphic to a quotient of a subspace of an ultraproduct of \(\theta\)-Hilbertian spaces for some \(\theta>0\) (see Corollary 6.7), where \(\theta\)-Hilbertian is meant in a slightly more general sense than in the author's earlier paper (1979).