Equilibrium conjecture for infinitely repeated games with incomplete information

Let $K$ be a finite set of states of nature, let $I$ and $J$ be finite sets of moves for two players, and let $R$ and $S$ be finite signal sets. A stochastic signalling function is a map

$$\Lambda\colon K\times I\times J\longrightarrow \Delta(R\times S),$$

where $\Delta(T)$ denotes the probability simplex on a finite set $T$. The corresponding infinitely repeated two-person, non-zero-sum game has incomplete information on one side, with the first player informed of the fixed state $k\in K$ and the second player receiving only the signals in $S$.

Equilibrium conjecture. For all signalling functions $\Lambda$, the corresponding infinitely repeated game of incomplete information on one side has an equilibrium, meaning a pair of strategies for which the limiting average expected payoffs exist in every state and neither player can improve the relevant limit superior by changing strategy.

This conjecture asks for existence of equilibrium for arbitrary signalling structures in infinitely repeated games with one-sided incomplete information. The supplied text gives no evidence that it has been resolved.