Is a batter's lost shoe detached offensive equipment?

[CecilOne]

Originally Posted by 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. ... snip ...

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?

USA & NFHS
If the ball is blocked, it is IDB, not DDB.

Removed incorrect opinion.
See below "any part of the uniform which inadvertently became detached during the ongoing play to be part of the field and not available for a blocked ball or int call "