Separation and divorce are two different things. They didn’t even separate. They were living together the whole time. It was all just a charade.
In NYS there is formal separation. There is a filing involved, however, so if they formally separated there should be paperwork.
Living together as in occupying the same residence, but perhaps not in the sense that phrase is typically used. Just like her mentor, Huma seems to have an arrangement with her spouse that falls outside of the common understanding of married life.