There's a group of players in the game that are truly all about the solo aspect of BST.
I, personally, love to solo and have been a solo artist for well into 7 years of my game play.
It seems if we allow our pets to fight mobs and heel them there's a good chance the mob will go unclaimed and if someone else wants that mob bad enough they'll take it from you.
Ie: King B was up and I threw a sheep at him. I got claim and was fighting him for a good 5-7 mins. I turned and put distance between my pet and myself and heeled my sheep to me so i could setup for a "Ready" and a jug switch.
There were two other people who walked up next to were I was and were watching. When I heeled my pet and pulled King B closer to me he went green and the Ninja snagged him within a second.
Another example, Serket, I got claim and there wasn't a person in the zone. I just let my pet fight it and got him down to 10% and decided to pull him close to a door so when he died I could just leave. When I heeled my pet and pulled Serket closer he went green and a party just had arrived and was leveling off beetles took claim on "accident" the leader said and killed the mob.
BST losing claim on mobs is very very common for those who solo. We spin mobs and use our pets in strat methods only to have them go green on random occasions costing us our time and our effort.
Another thing I would love to see is maybe a "Pet Focus" for our pets so when we send them after a mob it actually stays on that mob rather then spinning around attack anything that hits it. I've had my Pet run off across the zone after an AoE from a NM that wasn't even in my party.