Non-crossing chord diagram characterization of phi^4 regions

Let $n$ be even. Place $n-2$ ordered points labeled $3,4,cldots,n$ on a line, and let $ctheta_{ab}$ denote a chord in a non-crossing perfect matching. For $x=(x_0,cldots,x_{n-3})cincmathbb{R}^{n-2}$, define

$$H(x)=csum_{a=0}^{n-3}x_a+2csum_{a=0}^{n-4}csum_{b=a+1}^{n-3}(-1)^{b-a}cmin(x_a,x_{a+1},cldots,x_b).$$

Non-crossing chord diagram conjecture. The regions contributing to $A_n^{\phi^4}$ are in bijection with the $C_{n/2-1}$ possible non-crossing chord diagrams. The region associated with a diagram satisfies $x_{a-3}=x_{b-3}$ for each chord $ctheta_{ab}$, and $x_{a-3}=x_{b-3}<x_{c-3}=x_{d-3}$ whenever $ctheta_{ab}$ surrounds $ctheta_{cd}$; equivalently, these regions are all solutions to $H(x)=0$.

The paper explicitly states that this conjecture is proved later in the paper, so its database status is solved.