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 $C(P_X,P_Y)$ be the set of couplings of $P_X$ and $P_Y$, and let $\mathcal{G}(P_X,P_Y)$ denote the maximal guessing coupling value. A coupling is deterministic when its second variable is a function of its first.

Deterministic coupling conjecture.

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

if and only if $\mathcal{G}(P_X,P_Y)=1$. Equivalently, there exists a deterministic coupling $P_{X^nY^n}\in C(P_X^n,P_Y^n)$ for which $Y^n$ is a function of $X^n$ if and only if there exists a deterministic coupling $P_{XY}\in C(P_X,P_Y)$ for which $Y$ is a function of $X$.

This asks whether deterministic couplings can arise only at the one-letter level, rather than appearing for product distributions without a corresponding one-letter coupling. The supplied context gives no resolution, so the conjecture remains open.