= Solution
The factorial tail provides the connection with the supplied <Borel summation> formula. Near the first switching line choose $\lambda=-z^3$ with $\arg\lambda=3(\arg z-\pi/3)$, so $\lambda$ is near the positive real axis and $\lambda^{-1/3}=e^{i\pi/3}/z$. The endpoint contribution through $N-1$ terms is
$$
P_N(z)=\frac{e^\lambda}{3\lambda\Gamma(2/3)}\sum_{r=0}^{N-1}\frac{\Gamma(r+2/3)}{\lambda^r}.
$$
Writing $r=N+p$ in its formal remainder, with $\gamma=N+2/3$, gives
$$
\frac{\lambda^{-1/3}}{3\Gamma(2/3)}\sum_{p\geq0}\frac{\Gamma(\gamma+p)e^\lambda}{\lambda^{p+\gamma}}.
$$
Thus the <Borel remainder for a cubic exponential integral>, using the contour just above the pole specified in the paper, is
$$
\boxed{G(z)-P_N(z)=\frac C z+\frac{e^{i\pi/3}}{3\Gamma(2/3)z}I(\lambda,\gamma).}
$$
The continuation is chosen from the sector below the switching line. The term $C/z$ fixes that continuation; an alternative lateral pole contour would shift it by the corresponding residue. This is why the contour prescription must accompany the divergent factorial series.
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