The obstacle-number graph-counting conjecture

Let $f(h,n)$ denote the number of $n$-vertex graphs with obstacle number at most $h$. The function $g(n)$ satisfies $g(n)\in O(n\log^2 n)$.

Obstacle-number graph-counting conjecture. The number of such graphs satisfies

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

An improved upper bound on the dependence of the counting function on $h$ would improve lower bounds for the worst-case obstacle number of graphs. The paper presents this as a conjectural direction; no resolution is given.