Dujmović–Morin conjecture on the number of graphs with bounded obstacle number

Given a graph $G$, let its obstacle number be the minimum number of faces in a straight-line drawing whose union intersects every non-edge, and let $f(h,n)$ denote the number of $n$-vertex graphs with obstacle number at most $h$. For positive integers $n$ and $h$, the conjecture is

$$f(h,n)\in 2^{g(n)\cdot o(h)},$$

where $g(n)\in O(n\log^2 n)$. Dujmović–Morin conjecture. For all positive integers $n$ and $h$, we have $f(h,n)\in 2^{g(n)\cdot o(h)}$, where $g(n)\in O(n\log^2 n)$. This conjecture concerns the growth of the number of graphs admitting drawings with bounded obstacle number, and would constrain how many graphs can have small obstacle number. The cited context gives lower bounds on the maximum obstacle number, but does not state that this conjecture has been resolved.