> Can someone here ELI5 how they can be absolutely confident of these distances?

The way I read it is that they're absolutely confident of their fantastic young age, that is, their redshift.

As you point out, the age-to-distance conversion isn't quite settled, but that doesn't really matter in this context I think, as younger will still mean further away.

So these galaxies will still be the furthest away (for now) even if the distance ladder gets adjusted "inward", ie closer to us.