Finite-torus approximation conjecture for the DLA tree

Let ${\sf TDLA}_0(n,{ \sf Torus}(n^c))$ denote the finite-graph DLA tree with $n$ vertices rooted at $0$, and let ${\bf dla}_n$ denote the standard DLA tree with $n$ vertices. The finite-torus DLA conjecture. There \exists $C\in(1/2,1)$ such that, for every $c>C$,

$$D_{\sf var}\left({\sf TDLA}_0(n,{ \sf Torus}(n^c)),{\bf dla}_n\right)\mathop{\longrightarrow}_{n\to+\infty}0,$$

where $D_{\sf var}$ is total variation distance. The conjecture asserts that a sufficiently large polynomial torus reproduces the law of the standard planar DLA tree; the source gives no proof or resolution.