Capacity sparsity conjecture for transient random-walk aggregation

Let $R$ be a transient random walk on $\mathbb{Z}$, and let $A_n$ be the aggregate after $n$ particles. Write $\operatorname{Cap}(A_n)$ for its capacity. Capacity sparsity conjecture. Almost surely,

$$\operatorname{Cap}(A_n)=o(n).$$

The conjecture concerns the capacity growth of aggregates generated by arbitrary transient random walks on $\mathbb{Z}$. The surrounding discussion gives no proof or counterexample, so its status remains open.