Kang–Kim–Liu conjecture on single-graph extremal exponents

Let $r$ be a rational number in $[1,2]$. For a graph $H$, let $ex(n,H)$ denote the maximum number of edges in an $n$-vertex graph containing no copy of $H$.

Kang–Kim–Liu conjecture. For every rational number $r\in[1,2]$, there exists a graph $H$ such that

$$ex(n,H)=\Theta(n^r).$$

Kang, Kim and Liu stated this as a slightly weaker version of Erdős's single-graph conjecture, replacing an asymptotic equivalent by a matching order of growth. The source gives no evidence that this weaker conjecture has been resolved.