High-dimensional Brownian motion conformal-dimension conjecture

Let $B$ be $n$-dimensional Brownian motion. High-dimensional Brownian motion conjecture. For all sufficiently large $n$, $B$ is not minimal almost surely; moreover, for all sufficiently large $n$, its conformal dimension is $1$ almost surely. The source motivates this by the transience of Brownian motion in dimensions $n\geq3$ and the greater flexibility to deform paths in higher dimensions; no resolution is given.