The universal upper-bound conjecture for 2-dimensional simplicial complexes

Let $X$ be any 2-dimensional simplicial complex. The topological Turán number is denoted by $\operatorname{ex}_{\hom}(n,X)$. Universal upper-bound conjecture. There is a constant $C=C(X)$ such that

$$\operatorname{ex}_{\hom}(n,X)\leq Cn^{5/2}.$$

The paper proves the exponent $8/3$ for every fixed $X$ and conjectures that the optimal universal exponent is $5/2$, matching the known order for surfaces.