Gaussian Wasserstein bound for instantaneous preferential-attachment triangle counts

Fix $m\geqslant2$ and $\delta>0$. Let $T_n$ be the number of triangles in the instantaneous variant of the linear preferential attachment model, with connections specified by the instantaneous-attachment rule. Let $Z\sim\operatorname{N}(0,1)$, and write $\operatorname{d}_{\mathcal W}$ for the Wasserstein distance. Gaussian Wasserstein bound conjecture. One has

$$\operatorname{d}_{\mathcal W}\left(\frac{T_n-\operatorname{\mathds{E}}T_n}{\sqrt{\operatorname{Var}(T_n)}},Z\right)\lesssim\frac{1}{\sqrt{\log n}}.$$

The conjecture gives a quantitative Gaussian approximation for triangle counts in the positive-offset instantaneous model. The surrounding discussion says that simulations are consistent with a Gaussian limit, while the precise limiting behavior and rate remain unproved.