Asymptotically deterministic coupling conjecture

Let $X$ and $Y$ be random variables with distributions $P_X$ and $P_Y$, and let $P_X^n$ and $P_Y^n$ denote their $n$-fold product distributions. Let $H(X)$ and $H(Y)$ be their entropies, and let $\mathcal{G}(P_X,P_Y)$ denote the maximal guessing coupling value.

Asymptotically deterministic coupling conjecture. Assume $H(X)=H(Y)$. Then

$$\mathcal{G}(P_X^n,P_Y^n)\rightarrow 1$$

if and only if $\mathcal{G}(P_X,P_Y)=1$; equivalently, $P_X$ and $P_Y$ have the same probability values.

The conjecture identifies when equal-entropy distributions admit couplings that become asymptotically deterministic under product amplification. It is proved in the paper when either $P_X$ or $P_Y$ is uniform, while the general case remains open.