Polynomial Hirsch conjecture for simplicial-complex diameter

Let $H_s(n,d)$ denote the maximum diameter of a $d$-dimensional simplicial complex on $n$ vertices. The simplicial-complex diameter conjecture. For every $d\geq 3$, there is a threshold $n_0(d)$ such that, for all $n\geq n_0(d)$,

$$H_s(n,d) = \left\lfloor \frac 1d \cdot \binom{n}{d} - \frac{d+1}{d} \right\rfloor.$$

This asserts that the trivial upper bound is asymptotically tight in every dimension $d\geq 3$ and would determine the maximum diameter for all sufficiently large numbers of vertices. The source says that the detailed proof of the lower-bound strategy appears in a separate manuscript, while the matching upper-bound conjecture remains open.