Conjecture on the third Erdős–Szekeres-type function at $(6,1)$

Let $h(n,k)$ be the least positive integer such that every planar set ${\cal X}$ in general position with at least $h(n,k)$ points contains $n$ points forming a convex $n$-gon whose interior contains at most $k$ other points of ${\cal X}$. Let $g(n)=h(n,0)$. Conjecture on $h(6,1)$.

$$h(6,1)=g(6)=17.$$

The paper records the bounds $17\le h(6,1)\le127$ and notes that the conjecture would immediately imply $h(6,1)=h(6,2)=h(6,3)=h(6,4)=h(6,5)=17$. Its status was not resolved in the supplied source.