eero providing signal to POE injector?

I am NOT trying to power the eero with POE.

I have 2 eeros (J010011) in my mesh network. One is wired to my router/switch in bridge mode and the other is not wired. It works remarkably well.

Can I use one port of the remote eero to attach a POE Injector?

I am NOT trying to power the eero with POE. I want to use an injector to run a POE camera, and just want the signal from the eero.

camera -> POE injector -> eero -> my network

This is in an area with power, but running ethernet would be a huge pain.