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Update stuck on downloading 100%

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I had an identical problem. Contacted Tesla and they re-sent download. That stuck at 100 too. Reboots did not help. Out of desperation I backed out of garage taking the car off WiFi. After about 200 feet of driving I got the announcement that latest software was ready to install. Have no idea what the problem was, or why my odd solution worked. Try it and see if it helps.
 
I had the same issue for this most recent update. It was stuck downloading at 100%. So I scheduled a service appointment. The night before the appointment the download restarted and installed with no further problems. This might be a coincidence, but I have heard of Tesla doing something on their end just before a service appointment for this sort of issue.
 
I had this occur at least 3 occasions. Sometimes it took hours to restart (download reverted back to 50%), sometimes it took much quicker. In the end I left it overnight even though I slightly moved the car. I woke to find 'connect to wifi to download' etc so went outside found it stuck at 95% and rebooted my wireless access point. It completed to 100% and quickly offered the install.

I concluded that something else going on at 100% that is more sensitive to wifi strength than main download.