Ahlswede–Han conjecture on capacity with rate-limited state information

Consider a state-dependent channel $p(y|x,s)$ with independent and identically distributed state $S^n$, where the receiver obtains rate-limited state information through a separate communication link of rate $R_0$. Let $\hat{S}$ be an auxiliary random variable generated from $S$ according to $p(\hat{s}|s)$. Ahlswede–Han conjecture. The capacity is

$$C(R_0)=\max I(X;Y|\hat{S}),$$

where the maximum is over joint distributions of the form $p(x)p(s)p(y|x,s)p(\hat{s}|s)$ satisfying

$$I(S;\hat{S}|Y)\le R_0,$$

with $|\hat{\mathcal{S}}|\le |\mathcal{S}|+1$. The paper states that its main theorem confirms this conjecture in the special case where the state is a deterministic function of the channel input and output; thus the conjecture is resolved for that special case, while the general claim is not established here.