= Solution
Use exactly the stipulated unit-sound-speed mode functions for this integral. Their late-time values are real and $u_k'{}^*(\tau)=Hk^2\tau e^{ik\tau}/\sqrt{4\epsilon k^3}$. Consequently
$$
\prod_i u_{k_i}(0)u_{k_i}'{}^*(\tau)
=\frac{H^6}{64\epsilon^3k_1k_2k_3}\tau^3e^{iK\tau},\qquad K=k_1+k_2+k_3>0.
$$
The vacuum contour lies above the negative real time axis, so $e^{iK\tau}$ decays at its past endpoint. With $s=iK+0^+$, repeated differentiation of $\int_{-\infty}^0e^{s\tau}d\tau=1/s$ gives
$$
\boxed{\int_{-\infty(1-i0)}^0\tau^2e^{iK\tau}d\tau=\frac{2}{(iK)^3}=\frac{2i}{K^3}.}
$$
Substitution gives \b[the bispectrum with consistent operator ordering and the stated Hamiltonian]:
$$
\boxed{B_\zeta(k_1,k_2,k_3)=\frac{3H^4}{8\epsilon^2}\frac{\mathcal F}{k_1k_2k_3K^3},}
$$
$$
\boxed{\langle\zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\rangle_c
=\frac{+3H^4\pi^3}{\epsilon^2}\frac{1-c_s^2}{c_s^2}\mathcal A
\frac{\delta^{(3)}(\sum_i\mathbf k_i)}{k_1k_2k_3K^3}.}
$$
The sign is positive in front of $\mathcal A$, contrary to the target's negative sign. The printed negative result is obtained algebraically if one uses the unstarred derivatives, $e^{-iK\tau}$, and the conjugate past contour, which gives $-2i/K^3$, while retaining $-2i$ outside. That combination is not the stated vacuum contour or the stated external-before-vertex contraction. A consistently conjugated representation must also conjugate the external prefactor and has the positive result above. The <unit-speed cubic time-derivative bispectrum> records these conventions explicitly.
The approximation treats $H$, $\epsilon$ and the interaction coefficients as constant, uses the <de Sitter approximation>, retains only the stipulated vertex at tree level, and takes $k_i|\tau_{\rm final}|\to0$ after doing the convergent early-time integral. The $i0$ prescription must be kept while imposing the past boundary condition; an arbitrary early-time cutoff leaves spurious oscillatory boundary terms. For a physical noncanonical quadratic action with $c_s\ne1$, the mode functions also change to $H(1+ic_sk\tau)e^{-ic_sk\tau}/\sqrt{4\epsilon c_sk^3}$. They were not supplied here; the displayed calculation is conditional on the paper's specified modes, rather than a full self-consistent noncanonical power-spectrum calculation.
With the stipulated power spectrum $P_\zeta(k)=H^2/(4\epsilon k^3)$, an <equilateral bispectrum configuration> gives $B(k,k,k)/P(k)^2=2\mathcal F/9$. For the local-style normalization evaluated only at that triangle, $B(k,k,k)=(18/5)f_{\rm NL}^{\rm effective}P(k)^2$, this means
$$
\boxed{f_{\rm NL}^{\rm effective}(k,k,k)=\frac5{81}\mathcal F.}
$$
This is a normalization convention for an equilateral amplitude, not a claim that the shape is local. In a <squeezed bispectrum configuration>, the ratio to the long-short power product is suppressed by $(k_L/k_S)^2$. Small sound speed or large higher-kinetic derivatives can make $|\mathcal F|$ large enough for potentially observable <primordial non-Gaussianity>, whereas $c_s\to1$ with finite $\mathcal A$ suppresses this vertex. Detectability is conditional on amplitudes, the other cubic interactions, and perturbative control; it cannot be asserted from the given constants alone.
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