Cascade-channel capacity conjecture for finite-state channels

Let $p(y(i),s(i+1)\mid x(i),s(i))$ be a state-dependent channel with finite input and output alphabets satisfying $|\mathcal X|=|\mathcal Y|<\infty$. For a channel with memory, let $C_0$ denote its zero-error capacity, let $C_0^m$ denote the zero-error capacity of the $m$-fold cascade channel, and define

$$C_0^{\inf}=\lim_{m\to\infty}C_0^m.$$

Cascade-channel capacity conjecture. If $|\mathcal S|<\infty$, so that the channel is finite-state, then the Shannon capacity of the $m$-fold cascade channel converges to $C_0^{\inf}$ as $m\to\infty$. Conversely, there exist channels with finite input and output alphabets and infinite state space $|\mathcal S|=\infty$ for which the Shannon capacity of the cascade channel converges, as $m\to\infty$, to a value strictly larger than $C_0^{\inf}$.

The conjecture distinguishes finite-state channels, where asymptotic cascade capacity is predicted to equal the limiting zero-error capacity, from channels with infinite memory, where a strict separation can occur. The supplied text gives no resolution, so the conjecture remains open.