So I read the abstract and the link you included, and now I understand (at a high level) what they did. Basically, they measured the areas of two black holes before a merger and the area of the combined black hole after the merger. They concluded with >95% certainty that the area before the merge was not bigger than the area after the merge.

I am still confused about the apparent conflict between this theorem of Hawking's and the idea of Hawking Radiation, though. Wikipedia says this:

Hawking radiation reduces the mass and rotational energy of black holes and is therefore also theorized to cause black hole evaporation. Because of this, black holes that do not gain mass through other means are expected to shrink and ultimately vanish.

And this says:

Black holes are not decaying because there's an infalling virtual particle carrying negative energy; that's another fantasy devised by Hawking to "save" his insufficient analogy. Instead, black holes are decaying, and losing mass over time, because the energy emitted by this Hawking radiation is slowly reducing the curvature of space in that region. Once enough time passes, and that duration is enormous for realistic black holes, they will have evaporated entirely.

So on one hand, we have Hawking's theorem saying that black holes' area cannot get smaller, but on the other hand we have him saying that they will eventually evaporate and vanish.

I guess there would be a way to reconcile this if I understood it in more depth, but for now, I remain confused.