Fuss–Catalan enumeration and construction of regions for $A^{\phi^p}_n$

Place $n-2$ points labeled $0,1,\ldots,n-3$ in increasing order on the real line, and let $H(x)$ be the function in the global Schwinger formula for $A^{\phi^p}_n$. For a non-crossing $(p-2)$-chord $\theta_{a_1,a_2,\ldots,a_{p-2}}$, associate the equalities $x_{a_1}=x_{a_2}=\cdots=x_{a_{p-2}}$; if it surrounds $\theta_{b_1,b_2,\ldots,b_{p-2}}$, impose the corresponding strict inequality between the two common values. Fuss–Catalan region conjecture. The regions contributing to $A^{\phi^p}_n$ are in bijection with the $\textrm{FC}_{(n-2)/(p-2)}(p-2,1)$ possible non-crossing $(p-2)$-chord diagrams, and the region associated with each diagram is obtained by these equalities and nesting inequalities. These regions are all the solutions to $H(x)=0$, and their contributions produce all the $\textrm{FC}_{(n-2)/(p-2)}(p-1,1)$ trees of $\phi^p$. The supplied text gives no evidence that this conjectural classification and enumeration has been resolved.