Spherical truncation incidence-growth conjecture

Let $\Lambda_N^{\mathrm{sph}}$ denote the spherical truncation and let $S^{\mathrm{sph}}(N)$ be its associated incidence quantity. Spherical truncation incidence-growth conjecture. There is a constant $c\ge 0$ such that

$$S^{\mathrm{sph}}(N)=\Theta\!\bigl(N^3(\log N)^c\bigr).$$

The numerical comparison suggests that spherical truncation has different incidence growth from cubic truncation; establishing the asserted asymptotic would require sharp lattice-point estimates in thin spherical shells.