 In-in bispectrum conjugation rule

With ${\zeta }_k(\tau )=u_k(\tau )a_k+u_k^{\ast }(\tau )a_{-k}^{†}$, the operator order external field then vertex gives $⟨{\zeta }_k(0){\zeta }_p^{\prime }(\tau )⟩=(2\pi {)}^3\delta (k+p)u_k(0)u_p^{\prime }{}^{\ast }(\tau )$. For positive-frequency $u\propto e^{-ik\tau }$, these vertex contractions contain $e^{+iK\tau }$ and converge on ${\tau }_i=-\infty (1-i0)$. Conjugating the contraction representation also conjugates the outside in-in prefactor. Mixing the two changes signs and defeats the vacuum prescription.