timing chain keeps dropping/derailing only with partner
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Based on the new photo, your timing chain needs checking. Its quite uncommon to see both chains on the right side of the bike, but not impossible.

Do both timing chainrings have the same number of teeth, or do pedals go out of phase over distance ?

Are the chainrings in line? You might need a keeper on the front chainring - essentially a front derailleur mech that doesn't move.

Check for timing chain and chainring wear too - if its worn/elongated then fit a new 1/8" timing chain. Leave it too long and the chainrings will wear out as well.

1/8" chain will last much longer too, and will slip less because it doesn't flex left and right as much as normal 3/32" chain.