The Banach-space characterization conjecture for perfect earth mover's correlation

Let $X$ and $Y$ be Banach-space-valued random variables, and let $f$ be a similarity transformation of the Banach space. The Banach-space perfect-correlation conjecture. The axiom characterizing perfect dependence should hold in the form

$$\mathop{\mathrm{eCor}}(X,Y)=1 \quad\Longleftrightarrow\quad Y=f(X)\text{ with probability }1.$$

The paper establishes weaker axioms for arbitrary metric spaces and conjectures the full dependence-axiom package under general conditions such as Banach-space-valued variables; the supplied text does not establish this equivalence.