Universality of bounded extension realization spaces

For a finite point configuration $P\subset\mathbb{R}^d$ and an integer $k\geq 0$, let $[P]_k$ be the set of point configurations $Q\in(\mathbb{R}^d)^X$ such that every chirotope of size $|X|+k$ realizable on top of $P$ is realizable on top of $Q$. A primary basic semi-algebraic set is a semi-algebraic set defined by finitely many polynomial equalities and strict inequalities. Universality conjecture. For every $0\leq k<\infty$ and every primary basic semi-algebraic set $S$, there exists a finite point configuration $P\subset\mathbb{R}^2$ such that $[P]_k$ is stably equivalent to $S$. This conjecture asks whether universality still holds when realizations are restricted by extensions of bounded size; it would contrast with the rigidity obtained when all extensions are considered. The supplied text gives no resolution status.