The asymptotic limit conjecture for neighborly boxes

A box in $\mathbb{R}^d$ is an axis-parallel $d$-dimensional cuboid. Two boxes are $k$-neighborly if their intersection has dimension strictly smaller than $d$ but at least $d-k$. Let $n(k,d)$ be the maximum size of a family of pairwise $k$-neighborly boxes in $\mathbb{R}^d$. For every fixed integer $k\geqslant 1$, let $\gamma_k$ be a real number. The asymptotic limit conjecture.

$$\lim_{d\rightarrow \infty}\frac{n(k,d)}{d^k}=\gamma_k.$$

The conjecture is known for $k=1$, where Zaks's result gives $\gamma_1=1$, but it is widely open for every $k\geqslant 2$. The conjecture concerns the existence of an asymptotic density for the maximum size of pairwise $k$-neighborly box families; it is tempting, but not asserted here, that $\gamma_k=1/k!$.