Intermediate-regime diameter conjecture for random spanning trees in random environment

Let $K_n$ be the complete graph on $n$ vertices, let $\omega$ be the edge-disorder environment, and let ${{\mathcal T}}^\omega_{n,\beta_n}$ be the random spanning tree in random environment in the setting of Theorem T:main. Write $\operatorname{diam}({{\mathcal T}}^\omega_{n,\beta_n})$ for its graph diameter, and let $\widehat{\mathbb{P}}$ denote the averaged law. Intermediate-regime diameter conjecture. With high probability under $\widehat{\mathbb{P}}$,

$$\operatorname{diam}({{\mathcal T}}^\omega_{n,\beta_n})=\begin{cases} n^{\frac{1}{2}+o(1)}, & \beta_n\leq Cn, \\ n^{\frac{1}{2}-\frac{\gamma}{2}+o(1)}, & \beta_n=n^{1+\gamma+o(1)},\quad 0<\gamma<\frac{1}{3}, \\ n^{\frac{1}{3}+o(1)}, & \beta_n\geq Cn^{\frac{4}{3}}. \end{cases}$$

It predicts the diameter throughout the low-, intermediate-, and high-disorder regimes, interpolating between uniform and minimum spanning trees; the intermediate regime is left open by the theorem discussed immediately beforehand.