Exponent conjecture for k-holes avoiding a (k+1)-hole

For each fixed integer $k\ge 6$, let $h_k(n)$ be the maximum number of $k$-holes determined by a planar point set in general position of size at most $n$ and containing no $(k+1)$-hole. A $k$-hole is a $k$-element subset in convex position whose convex hull contains no other point of the set.

Exponent conjecture for $h_k(n)$. For every fixed integer $k\ge 6$,

$$h_k(n)=O_k\!\left(n^{\lceil k/3 \rceil}\right).$$

The paper proves upper and lower bounds with exponents $\lceil k/2\rceil$ and $\lfloor k/3\rfloor$, respectively, and conjectures that the true exponent is closer to the lower bound. Horton sets and grid Horton sets provide supporting examples, but the asserted upper bound remains open.