Regular shooting takes place in the first week of Olympics so I can't see MP getting the early love. What works against this sport as well is two of the biggest SOG powerhouses (USA & China) are both non-factors so there is no political pressure to feature MP in a highlighted spot on the schedule. Running it towards the ends when team sports medal rounds/finals take place puts this firmly in the graveyard of Olympic schedule for sure.
Funnily enough the last time MP was held in the first week was 1992 Barcelona where Poland won two gold medals and the dreaded equestrian was the last event of the five (the competition lasted four days, impossible in the current day & age).
Back then athletes had a training round available, not the 15 minutes "get to know the horse" we have now but like I've said, the competition wasn't held during a single day so they had time to accomodate the horses.