Spreer–Tobin's sharp vertex bound conjecture in arbitrary dimensions

Let $\mathcal{T}$ be a triangulation of a closed and connected $d$-dimensional manifold, with $d>0$, $f_0$ vertices, and $f_d$ facets. Spreer–Tobin's sharp vertex bound conjecture. The following hold: (a)

$$f_0 \leq \begin{cases} \frac{f_d}{2}+d & \text{if } d \text{ is even, or if } d \text{ is odd and } f_d<d \text{ and } f_d \text{ is even},\\ f_d+\frac{d-1}{2} & \text{otherwise};\end{cases}$$

(b) for every feasible value of $f_d$ in every dimension, there exists an $f_d$-facet triangulation of the $d$-sphere attaining equality in the applicable bound. This conjecture seeks sharp dimension-dependent vertex bounds, with the even-dimensional case and the odd-dimensional small odd-facet case identified as unresolved in the paper.