The causal state-information uselessness conjecture for binary-input channels

Let $W$ be a channel from $[\![1,2]\!]$ to $[\![1,n]\!]$, and let $\mathcal{S}$ be the state space. Suppose

$$W=\sum_{s\in\mathcal{S}}p_S(s)K^{(s)},$$

where $S$ denotes the channel state. Causal state-information uselessness conjecture. If, for every $1\leq j\leq n$, the two entries $K^{(s)}_{1,j}$ and $K^{(s)}_{2,j}$ have an order—either $K^{(s)}_{1,j}\leq K^{(s)}_{2,j}$ for all $s$, or $K^{(s)}_{1,j}\geq K^{(s)}_{2,j}$ for all $s$—independent of $s$, then causal state information available at the encoder cannot increase the capacity of $W$.