Interpolation between projective and injective spaces

Question

Suppose $(\Omega,\mu)$ be a $\sigma$-finite measure space. Suppose $X$ is a Banach space and $L_p(\Omega;X)$ be the corresponding Bochner space for $0<p\leq\infty.$ Is it true that the complex interpolaion $(L_1(\Omega)\widehat{\otimes}X,L_\infty(\Omega)\check{{\otimes}}X)_{1/p}$ is isometrically isomorphic to $L_p(\Omega;X)$ for $1<p<\infty.$ Here $\widehat{\otimes}$ and $\check{{\otimes}}$ are projective and injective tensor products of Banach spaces respectively.