I would not put my dog at risk for thew sake of a play session. Never, having heard of it, I looked it up on the net and found an interesting site mentioning an outbreak at a Guidedogs for the Blind centre.
The cyst cells are expelled in the faeces and can stay active for months. Just wiping a dog's backside will not remove any cysts caught up in its coat and cleaning up poo could leave cysts on the ground, especially in grass. Also,the other dogs could be carriers, even if they have no symptoms.
The solution given on the website was medical traatment and isolation of the dogs infected, together with the use of special disinfectants. this may need to go on for several weeks.
It seems that infection is a real possibility, while the dog is still suffering and even for a while afterwards. It appears that not only your dog, but you, too, could get infected.
Barbara, Monty, Joly and Teddy.