Comparison of eccentricity constraints between Cadieux et al. (2024a) (in blue; C. Cadieux, priv. comm.) and this work (in orange). The posterior from Cadieux et al. (2024a) was used as the prior for our eccentricity parameters. The additional constraint from the eclipse time significantly narrows the range of $\sqrt{e}\cos \omega$ (top left), which helps tighten the eccentricity constraint much closer to zero (top right). In the bottom left plot, we see that the new joint-posterior of $\sqrt{e}\cos \omega$ and $\sqrt{e}\sin \omega$ is largely constrained to lie between values which correspond to 1 to 5 minute early eclipses (relative to a circular orbit). The constraints on $\sqrt{e}\sin \omega$ are mostly unchanged from the prior values however (bottom right).