otterbox+iphone+stylus+gloves= not working

The touch screen doesn't work by pressure so you can press as hard as you want with the stylus but nothing will happen. The screen responds to the static electricity from skin contact. My guess is that the stylus acts as an extension of you finger but having it on the carpet is screwing that up.... It also explains why wearing gloves renders the stylus useless. If I had to guess, if you read the packaging for the stylus it explains how it works.

You're point of entry for this problem should probably be POGO, not OtterBox. The OtterBox Defender case is made of plastic and silicon, and is non-conductive, so I would rule out the requirement to be holding the cased iPhone in your hand. Try placing the cased iPhone on another non-conductive surface (like a wooden table), and touch it with the POGO. Observe what happens.

As john.grayson stated, I would suspect POGO has detailed how their device works either on the packaging on on their website, since the iPhone screen uses your finger's conductivity to work.

The issue with the POGO not working when placed on the carpet may be related to the static electricity build-up in the carpet interfering with the POGO's functionality. The conductivity path should be from your hand to the POGO. Holding the iPhone (or not) won't have any effect. Again this can be tested by placing your (cased or uncased) iPhone on a non-conductive surface (and not touching it), and experimenting with the POGO (no gloves, gloves, etc) and see if the there is any change.

I suspect you will find the issue solved when you have skin-to-POGO contact, regardless of where the iPhone is placed (with some exceptions like sitting on the carpet as per above) or whether it is in the Defender or uncased.

The display of an iPhone doesn't work with pressure but with conduction of your body. This all has to do with physics, so actually if you know something about this you can figure this out.