Critical logarithmic central limit conjecture for the on-line nearest-neighbour graph

Let $\d835𝒪^{d,α}(\d835𝒰_n)$ be the total power-weighted edge length of the on-line nearest-neighbour graph on $n$ uniform points in the unit cube, and let $\d835𝒩(0,σ^2)$ denote the normal distribution with mean zero and variance $σ^2$. Write $\d835\stackrel{d}{\longrightarrow}$ for convergence in distribution.

Critical logarithmic CLT conjecture. Let $d\in\mathbb{N}$. There exists a constant $σ_{d,d/2}^2\in(0,\infty)$ such that

$$(\log n)^{-1/2}\tilde{𝒪}^{d,d/2}(𝒰_n)\stackrel{d}{\longrightarrow}𝒩(0,σ_{d,d/2}^2),\qquad n\to\infty.$$

The conjecture concerns the critical exponent $α=d/2$, which the surrounding results identify as exceptional. In particular, it predicts Gaussian fluctuations with logarithmic rather than polynomial normalization; the supplied text gives no resolution.