Average number of realizable one-point extensions of planar chirotopes

Let $n\geq 3$, and choose uniformly at random a planar realizable chirotope $\chi$ of size $n$. For each realization of $\chi$, count the realizable one-point extensions, and average this count over all realizations of $\chi$. Extension-counting conjecture. There exists a constant $c$ such that this average number of realizable one-point extensions is at most

$$c\cdot n^4.$$

The conjecture expresses the expectation that a typical realizable order type has its one-point extensions accessible from a common realization at the same polynomial scale as the upper bound for any fixed realization. The supplied text gives no resolution status.