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It's normal for the swingarm to get hot, it becomes even longer and the chain tension is up to 5mm less
the active key works without battery also as passive directly at the antenna
for model year 2010-2012 there is a Handsfree update which bypasses the 2 limit switches. Ducati RS SAT 876
https://static.nhtsa.gov/odi/tsbs/2017/MC-10110349-9999.pdf
https://static.nhtsa.gov/odi/tsbs/2018/MC-10143632-9999.pdf
https://static.nhtsa.gov/odi/rcl/2019/RCSB-19V382-9239.pdf