Asymptotic optimality of the alternative payoff for parallel fingerprinting channels

Let $L$ be the number of TV channels, let $k$ be the coalition size, and let $\tilde{L}=1$. Write $J_{k,L}$ for the original payoff and define the alternative payoff

$$R_{k,L}=J_{k,L}-\frac{1}{k}I(\hat Y;\hat S\mid\hat X_{\mathcal K}).$$

Here $\hat Y$ is the vector of pirate outputs, $\hat S$ records the assignment of pirates to the TV channels, and $\hat X_{\mathcal K}$ is the part of the code matrices accessible to the coalition. Asymptotic optimality conjecture. In the binary alphabet case,

$$\max_{f_W}\min_{\{\pi\},p_{\hat C}}R_{k,L}\longrightarrow\frac{L^2}{k^2 2\ln 2},$$

with optimal strategies $f_W^*(w)=(\pi\sqrt{w(1-w)})^{-1}$, $g^*(w)=w$, and

$$p_{\hat C}^*(\hat c)=\prod_{l=1}^{L}\delta(c^l,k/L).$$

The conjecture asserts that the alternative payoff has the same asymptotic maximin value as the original multiple-channel payoff. The proposed strategies are the arcsine bias distribution, the interleaving attack, and an even allocation of pirates across channels; the source provides motivation but no resolution.