As far as I understand, that paper talks only about MALL, which is a small propositional fragment of LL's, in particular, exponentials are missing. Extending models for small fragments to full LL has always been difficult. Note that the Lincoln et al paper is > 2 decades old. Has anyone extended it to full LL?
Note also that this paper doesn't seem to be talking about Minimax.
Note also that this paper doesn't seem to be talking about Minimax.