Matching lower-bound conjecture for the universal exponent of sphere homeomorphs

Let $d$ be a positive integer, and let $\lambda_d$ be the universal exponent for homeomorphs of fixed $d$-complexes, defined by

$$\operatorname{ex}(n,\mathcal{H}(S))\leq O(n^{d+1-\lambda_d})$$

for every fixed $d$-complex $S$. Matching lower-bound conjecture. The universal exponent satisfies

$$\lambda_d\geq\frac{d+1}{2^{d+1}-2}.$$

The paper proves the corresponding upper restriction under exponential growth of the family of simplicial $d$-spheres, while Long, Narayanan, and Yap provide a general lower bound; whether this sharper bound holds remains open.