Newman–Pavelka sphere exponent conjecture

Let $d\geq 3$, let $\mathbb S^d$ be the $d$-dimensional sphere, and let $\operatorname{ex}_{\hom}(n,\mathbb S^d)$ denote its topological Turán number. Newman–Pavelka sphere exponent conjecture. It holds that

$$\operatorname{ex}_{\hom}(n,\mathbb S^d)=O\left(n^{d+1-(d+1)/(2^{d+1}-2)}\right).$$

This would improve the known Cauchy–Schwarz upper bound $O(n^{d+1-2^{1-d}})$. The source notes that the conjectured exponent is tight for several structured families of triangulations, while the general case remains open.