Dirichlet distribution conjecture for asymptotic sector widths of the radial spanning tree

Let $m$ be the number of unbounded trees, and condition on $m\in\{2,3,4,5\}$. For cyclically labeled interfaces, define the sector widths

$$\phi(i+1):=\theta(i+1,i+2)-\theta(i,i+1),$$

with the conventions $\theta(m,m+1)=\theta(m,1)$ and $\theta(m+1,m+2)=\theta(1,2)$. Thus $(\phi(1),\dots,\phi(m))$ records the asymptotic widths of the unbounded trees and has total width $2\pi$. Dirichlet distribution conjecture. Conditionally on $m\in\{2,3,4,5\}$, the vector $(\phi(1),\dots,\phi(m))$ has a distribution close to a symmetric Dirichlet distribution of order $m$ on $[0,2\pi)$ with parameter $\alpha\ne 1$, denoted $\operatorname{Dir}(m,[0,2\pi),\alpha)$. This conjecture proposes a non-independent model for the interface directions, consistent with the preceding result that each sector has expectation $2\pi/m$; the precise meaning of “close” and the value of $\alpha$ remain unspecified.