Strict inclusion conjecture for causal information integration and causal information structure

Let $\mathcal{M}_{CII}$ denote the model of causal information integration and let $\mathcal{M}_{CIS}$ denote the model of causal information structure, with $\mathcal{M}_{CII}\subseteq\mathcal{M}_{CIS}$. Approximation is understood with respect to the distribution space under consideration. Strict inclusion conjecture. It is not possible to approximate every distribution $Q\in\mathcal{M}_{CIS}$ with arbitrary accuracy by an element $P\in\mathcal{M}_{CII}$. Therefore,

$$\mathcal{M}_{CII}\subsetneq\mathcal{M}_{CIS}.$$

The conjecture would show that allowing the state space of $W$ to grow does not make the causal information integration model coincide with the causal information structure model. It is based on numerically calculated examples, and no resolution is given here.