Bruno's non-crossing chord diagram conjecture for phi^4 regions

Let $n$ be even, let $x=(x_0,cldots,x_{n-3})cincmathbb{R}^{n-2}$, and define

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

A non-crossing chord diagram is a perfect matching of $n-2$ ordered points, with chords $ctheta_{ab}$ drawn without crossings. Bruno's conjecture. The regions contributing to $A_n^{\phi^4}$, namely the regions where $H(x)=0$, are in bijection with the set of all $C_{n/2-1}$ non-crossing chord diagrams. For the diagram containing $ctheta_{ab}$ and, nested inside it, $ctheta_{cd}$, the corresponding region is specified by $x_a=x_b$ and $x_a=x_b<x_c=x_d$.

These regions are precisely the polyhedral cones that contribute to the global Schwinger formula for the phi^4 amplitude. The source gives the conjecture as an aim of the paper; its resolution is not specified here.