Asymptotic binary fingerprinting capacity for multiple TV channels

Let $k$ be the coalition size, let $L$ be the number of parallel TV channels, and let $C_{\rm fp}^{\rm binary}(k,L)$ denote the binary fingerprinting capacity in this setting. Assume $\tilde{L}=1$. Multiple-channel capacity conjecture. The capacity satisfies

$$\max_{f_W}\min_{\{\pi\},p_{\hat C}}C_{\rm fp}^{\rm binary}(k,L)\longrightarrow\frac{L^2}{k^2 2\ln 2},$$

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

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

This conjecture extends the single-TV-channel fingerprinting-capacity asymptotic to parallel channels. The source motivates it by reducing the problem, after conditioning on the channel assignment, to independent single-channel problems, but gives no proof for general $L$.