The conjectured lower bound for the centre-of-mass escape rate

Let $X$ be the increment random variable of the random walk in $\mathbb{R}^d$, and let $G_n$ denote its centre of mass after $n$ steps. Assume that the support of $X$ is not contained in a one-dimensional subspace of $\mathbb{R}^d$. Escape-rate conjecture. Almost surely,

$$\liminf_{n \to \infty} \frac{\log \|G_n\|}{\log n} \geq \frac{1}{2}.$$

Obtaining necessary and sufficient conditions for recurrence and transience of $G_n$ is stated to be an open problem; this conjecture asserts that in dimensions at least two the centre of mass is at least as transient as in the established diffusive-rate theorem.