Conjecture on all occupation times of unit-ball surfaces

Let $d\geq 3$, let $\mathcal{S}(1)=\{\mathbf{z}\in\mathcal{Z}_d:\|\mathbf{z}\|=1\}$, and define the occupation time of the unit-ball surface centered at $\mathbf{z}$ by

$$\Xi(\mathbf{z},n):=\Xi(\mathcal{S}(1)+\mathbf{z},n).$$

Writing $p=\mathbf{P}_{\mathbf{e}_1}(T_{\mathcal{S}(1)}<T)$ and $\kappa=1/[-\log(p+1/(2d))]$, the occupation-time conjecture. For any $\varepsilon>0$ with probability $1$ there exists a random variable $n_0=n_0(\varepsilon)$ such that if $n\geq n_0$ then for all $k=1,2,\dots,\lfloor(1-\varepsilon)\kappa\log n\rfloor$ there exists $\mathbf{z}\in\mathcal{Z}_d$ such that

$$\Xi(\mathbf{z},n)=k.$$

The preceding result identifies $\kappa$ as the almost-sure asymptotic scale of the maximum occupation time, and this conjecture predicts that every smaller positive occupation level occurs somewhere. The statement is presented as a suggested consequence of the known maximum asymptotics; its resolution is not supplied here.