The universal exponent asymptotic conjecture

For each dimension $d$, let $\epsilon_d$ denote the universal exponent parameter for $d$-dimensional simplicial complexes. Universal exponent asymptotic conjecture. The universal exponent satisfies

$$\epsilon_d=2^{-d-o(d)}.$$

The source records the known lower bound $\epsilon_d\geq d^{-2d^2}$ and suggests that the universal exponent should behave similarly to the exponent for the $d$-dimensional sphere. The proposed asymptotic remains open.