We show that, if a a finite-dimensional operator space E is such that X contains E C-completely isomorphically whenever X** contains E completely isometrically, then E is $2^{15}{C}^{11}$-completely isomorphic to Rₘ ⊕ Cₙ for some n, m ∈ ℕ ∪ 0. The converse is also true: if X** contains Rₘ ⊕ Cₙ λ-completely isomorphically, then X contains Rₘ ⊕ Cₙ (2λ + ε)-completely isomorphically for any ε > 0.