Coupling Brownian loop soups and random walk loop soups at all polynomial scales

Lawler and Trujillo Ferreras constructed a well-known coupling between the Brownian loop soups on $\mathbb{R}^2$ and the (discrete-time) random walk loop soups on $\mathbb{Z}^2$ (one rescales the random walk loops by $1/N$, their time parametrizations by $1/(2N^2)$, and lets $N\to \infty$), which le