The logarithmic-correction conjecture for loop-erased random walk in dimension four

Let $L$ be a loop-erased random walk on the four-dimensional $(N,4)$-torus. Its mean-field scale is $N^2$.

Four-dimensional logarithmic-correction conjecture. The accurate upper-bound scale for the length of $L$ is

$$N^2\log^{1/6}N.$$

The paper proves an upper bound of the form $N^{2+\epsilon}$ for every $\epsilon>0$ and reports evidence for the sharper logarithmic correction above. Determining this precise correction remains open in the supplied source.