= In-in bispectrum conjugation rule
{title2=$\langle\zeta(0)\zeta^{\prime}(\tau)\rangle\propto u(0)u^{\prime *}(\tau)$}
With $\zeta_k(\tau)=u_k(\tau)a_k+u_k^*(\tau)a^\dagger_{-k}$, the operator order external field then vertex gives $\langle\zeta_k(0)\zeta_p'(\tau)\rangle=(2\pi)^3\delta(k+p)u_k(0)u_p'{}^*(\tau)$. 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.
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