Transverse lattice-point counting conjecture for spherical bands

Let $A_{k,d,\lambda}$ denote the maximal number of lattice points in the intersection of $k$ unit-width, $\nu$-transverse bands on the sphere $\lambda S^{d-1}$, with $1\le k\le d-1$. Spherical-band counting conjecture. For $d\ge2$ and $1\le k\le d-1$,

$$A_{k,d,\lambda}\lesssim1,\qquad\text{when }d-k=1,$$

and

$$A_{k,d,\lambda}\lesssim_\varepsilon \lambda^{d-k-2+\varepsilon},\qquad\forall\varepsilon>0,\quad\text{when }d-k\ge2.$$

The paper proves lower bounds of the expected order in the relevant ranges, but the corresponding upper bounds are presented as conjectural.