Every math-based prediction about the real world will be based on two parts:
1) The world behaves like this model [up to our measurement accuracy].
2) The model implies this prediction.
Their claim would only be speaking to 2) -- that, under the model, you cannot reach absolute zero. The model can still be wrong.
I would argue (as I did in a sibling thread https://news.ycombinator.com/item?id=13878107) that it is this that can't happen. Mathematics has no power over the real world, only over mathematical models (optionally of the real world). A proof of mathematical impossibility does not constrain the real world; it only allows us to deduce that, if the real world does the mathematically impossible, then our models are wrong (or at least incomplete).
Mathematical proof is mathematical. It has nothing to do with the physical world.
P.S.: Note that 'mathematically shown to be physically impossible' still allows the possibility that the mathematical model was invalid. E.g., what if, after you have mathematically shown it to be physically impossible, you conduct an experiment and the "physically impossible" happens? That means you just provided an evidence that the mathematical model is invalid.
P.S.: There are tons of resources on this topic [1]. Read and make up your own mind.
[1] https://www.google.com/search?q=mathematical+proof+of+physic...