With any number of players, leads will automatically disappear once a certain amount of blocks away. This is an issue for all mobs, and horses in particular because they cannot sit and therefor will not stay.
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duplicates
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This is not a duplicate. Leads disappear on server restart, which is MC-14826. Leads also disappear on chunk unload/reload. I had 5 horses tethered to fenceposts, and after the chunks unloaded then reloaded, four of the leads were gone.
Duplicate of MC-14826, please use the search function to see if your bug has already been submitted. Currently over 55% of tickets are being closed as duplicate.