Is a batter's lost shoe detached offensive equipment?

[teebob21]

R1 on first, no outs. BR comes to the plate with an untied shoe. BR hits a soft base hit to left-center. While running to first, she loses the untied shoe between first and home.

BR rounds the bag at 1B aggressively and F8 fields the ball. F8 overthrows F3 at 1B while R1 is advancing to 3B. The ball hits the dugout fencing and rebounds off, hitting the shoe, causing F2 to chase the ball down in foul territory off the shoe deflection. (*) R1 rounds 3B and scores following a throw from F2 to F1 covering; single-shoe BR reaches 2B but is put out at 3B.

Third World Warning. This play actually happened, although in my game, there was no play at home or 3B. Everything after the (*) is hypothetical.

Is the shoe detached offensive equipment? Does play stand or are we playing under a delayed dead ball after the ball hits the shoe? If DDB, where do we place the runners and are there any outs? Is there a difference in this scenario between USA/Fed/NCAA?