Proposition 3.6 - uniqueness on paired routes

For every pair of probability measures $\mu,\nu$ on $\mathbb R$ that are mutually singular, define $F(x)=\operatorname{toReal}(\mu((-\infty,x]))-\operatorname{toReal}(\nu((-\infty,x]))$. For $p=(a,b)$, call $p$ a positive point of $F$ when $b$ lies in the unordered closed interval between the left limit of $F$ at $a$ and $F(a)$, and there exists $\varepsilon>0$ such that every $q=(c,d)$ satisfying $d$ in the unordered closed interval between the left limit of $F$ at $c$ and $F(c)$, $|c-a|<\varepsilon$, and $c\ne a$, also satisfies $0<(d-b)(c-a)$; call $p$ a negative point when the same conditions hold with $(d-b)(c-a)<0$. A real number $h$ is a regular level when there exist $n\in\mathbb N$, including possibly $n=0$, and a strictly increasing family $(x_i)_{i\in\{0,\ldots,2n-1\}}$ such that the set of all $x\in\mathbb R$ for which $h$ lies in the unordered closed interval between the left limit of $F$ at $x$ and $F(x)$ is exactly $\{x_i:0\le i<2n\}$; every $(x_i,h)$ is either a positive or a negative point; and, for every $i$, $(x_i,h)$ is a positive point if and only if $(0<h$ if and only if $i$ is even$)$. Let $P\subseteq\mathbb R^2$ consist of all $(a,b)$ for which there exists $h\ne0$ such that $h$ is a regular level, $(a,h)$ is a positive point, $(b,h)$ is a negative point, and either $h>0$, $a<b$, and no $z\in(a,b)$ has $h$ between the left limit of $F$ at $z$ and $F(z)$, or $h<0$, $b<a$, and no $z\in(b,a)$ has that property. Then, for any measures $\pi,\pi'$ on $\mathbb R\times\mathbb R$, if both first-coordinate pushforwards equal $\mu$, both second-coordinate pushforwards equal $\nu$, and both measures assign mass zero to $P^{\mathsf c}$, one has $\pi'=\pi$.