The trans-neptunian object (58534) 1997 CQ$_{29}$ (a.k.a. Logos) is a
resolved wide binary in the dynamically Cold Classical population. With Hubble
Space Telescope resolved observations where the primary Logos is well separated
from its secondary Zoe it can be established that Logos has a time-variable
brightness. Logos’ brightness varied by several tenths of a magnitude over a
short timescale of hours while the brightness variability of Zoe was on a
longer timescale. New unresolved ground-based observations obtained with the
Lowell Discovery Telescope and the Magellan-Baade telescope confirm at least
one highly variable component in this system. With our ground-based
observations and photometric constraints from space-based observations, we
suggest that the primary Logos is likely a close/contact binary whose
rotational period is 17.43$\pm$0.06 h for a lightcurve amplitude of
0.70$\pm$0.07 mag while Zoe is potentially a (very) slow rotator with an
unknown shape. Using the Candela software, we model the Logos-Zoe system and
predict its upcoming mutual events season using rotational, physical, E
mutual orbit parameters derived in this work or already published. Zoe’s shape
and rotational period are still uncertain, so we consider various options to
better understand Zoe. The upcoming mutual event season for Logos-Zoe starts in
2026 and will last for four years with up to two events per year. Observations
of these mutual events will allow us to significantly improve the physical and
rotational properties of both Logos and Zoe.

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