Possible polarization dichotomy for modified mutual information quantities

Let $\mathcal{A}'_n i\text{ and }\mathcal{S}_n i$ be the index sets defined in the construction, with $\mathcal{S}_n\tu001csubseteqmathcal{A}_n^{text{c}}$, and let $\mathcal{D}_n=\mathcal{A}'_n\tu001ccupmathcal{S}_n$. For $i i\tu001cmathcal{D}_n$, define

$$J_i=I(W_i;\mathbf{W}_{\mathcal{D}^{(i)}_n},\mathbf{W}_{\mathcal{D}^{\text{c}}_n},\mathbf{Y}).$$

Possible polarization dichotomy. Fix a $0<\delta<1$. There exists a partition of $\mathcal{S}_n$ into two sets $\mathcal{S}'_n$ and $\mathcal{S}”_n=\mathcal{S}_n\tu001csetminusmathcal{S}'_n$ such that, for sufficiently large $n$,

$$J_i<\delta\text{ for all }i imathcal{S}'_n,$$

and

$$J_i>1-\delta\text{ for all }i imathcal{S}”_n.$$

This conjecture concerns the unresolved polarization behavior of the indices in $\mathcal{S}_n$; the corresponding assertion for the good indices in $\mathcal{A}'_n$ is established by the preceding lemma.