There are two follow up papers showing the representations are "entangled", a euphemism for statistical garbage, but I can't be bothered at the moment to find them.
However the whole issue of othello is a nonsequiteur which indicates that people involved here don't really seem to understand the issue, or what a world model is.
A "world model" is a model of a data generating process which isn't reducible-to or constituted by its measures. Ie., we are concerned for the case where there's a measurement space (eg., that of the height of mercury in a thermometer) and a target property space (eg., that of the temperature of the coffee). So that there is gap between the data-as-measure and its causes. In language this gap is massive: the cause of my saying, "I'm hungry" may have nothing to do with my hunger, even if it often does. For "scientific measuring devices", these are constructed to minimize this gap as much as possible.
In any case, with board games and other mathematical objects, there is no gap. The data is the game. The "board state" is an abstract object constituted by all possible board states. The game "is made out of" its realisations.
However the world isnt made out of language, nor coffee made out of thermometers. So a model of the data isnt a mdoel of its generating process.
So whether an interpolation of board states "fully characterises", someway, an abstract mathematical object "the game" is so irrelevant to the question it betrays a fundamental lack of understanding of even what's at issue.
No one is arguing that a structured interpolative model (ie., one given an inductive bias by an NN architecture) doesn't express properties of the underlying domain in its structure. The question is what happens to this model of the data when you have the same data generating process, but you arent in the interpolated region.
This problem is, in the limit of large data, impossible for abstract games by their nature, eg., a model classifying the input X into legal/illegal board states is the game.
Another way of phrasing this is that in ML/AI textbooks often begin by assuming there's a function you're approximating. But in the vast majority of cases where NNs are used, there is no such function -- there is no function tokens -> meanings (eg., "i am hungry" is ambigious).
But in the abstract math case there is a function, {boards} -> Legal|Illegal is a function, there are no ambiguous boards
So: of the infinite number of f* approximations to f_game, any is valid in the limit len(X) -> inf. Of the infinite number f*_lang to f_language, all are invalid (each in their own way).