They could potentially watermark the model in order to identify the output. There are techniques for doing that, for example by randomly assigning token into groups A and B, group A probability is increased over group B, if group A is over-represented, chances are that that the output comes from the watermarked model.

How effective these techniques are and how acceptable as a proof it is is yet to be defined.

I don't think it is the case here, they probably don't really care, and watermarking has a cost.