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Systems with ultra-short-period planets (USPs) tend to possess larger mutualinclinations compared to those with planets located farther from their hoststars. This could be explained due to precession caused by stellar oblatenessat early times when the host star was rapidly spinning. However, stellaroblateness reduces over time due to the decrease in the stellar rotation rate,and this may further shape the planetary mutual inclinations. In this work, weinvestigate in detail how the final mutual inclination varies under the effectof a decreasing $J_2$. We find that different initial parameters (e.g., themagnitude of $J_2$ and planetary inclinations) will contribute to differentfinal mutual inclinations, providing a constraint on the formation mechanismsof USPs. In general, if the inner planets start in the same plane as thestellar equator (or co-planar while misaligned with the stellar spin-axis), themutual inclination decreases (or increases then decreases) over time due to thedecay of the $J_2$ moment. This is because the inner orbit typically possessesless orbital angular momentum than the outer ones. However, if the outer planetis initially aligned with the stellar spin while the inner one is misaligned,the mutual inclination nearly stays the same. Overall, our results suggest thateither the USP planets formed early and acquired significant inclinations(e.g., $\gtrsim30^\circ$ with its companion or $\gtrsim10^\circ$ with its hoststar spin-axis for Kepler-653c) or they formed late ($\gtrsim$Gyr) when theirhost stars rotate slower.

Mutual Inclination of Ultra-Short-Period Planets with Time Varying Stellar J2-moment