Kalai's conjecture on the abundance of neighborly simplicial spheres

Let $s(d,n)$ denote the number of simplicial $(d-1)$-spheres with $n$ labeled vertices, and let $\operatorname{sn}(d,n)$ denote the number of $\lfloor d/2\rfloor$-neighborly simplicial $(d-1)$-spheres with $n$ labeled vertices. Kalai's conjecture. For all $d\geq 4$,

$$\lim_{n\to\infty}\frac{\log \operatorname{sn}(d,n)}{\log s(d,n)}=1.$$

The conjecture asserts that neighborly simplicial spheres account asymptotically for the logarithmic growth of all simplicial spheres. It is motivated by Shemer's sewing construction and Kalai's proposed abundance principle; the supplied source does not indicate a resolution.