Ambient cubic-container problem for visibility cliques

Let $A\subset\mathbb{R}^2$ be a finite point set. A visibility patch is a subset $W\subset\Gamma(\mathbb{R})$ of a real cubic on which the relevant pairs of points are considered, and an ambient blocker for a pair from $A\cap W$ is a point of $A\setminus\Gamma$ lying in the open segment joining that pair. Ambient cubic-container problem. For every $k,\ell$ there are constants $\alpha>0$ and $\beta<1/(\ell-1)$ such that, if $A\subset\mathbb{R}^2$ has $n$ points, no $k$ collinear points, and no $\ell$ mutually visible points, then there is a real cubic $\Gamma$ and a visibility patch $W\subset\Gamma(\mathbb{R})$ for which

$$|A\cap W|\ge \alpha n$$

and the number of ambient blockers in $A\setminus\Gamma$ for pairs from $A\cap W$ is at most

$$\beta |A\cap W|.$$

This is identified as the missing ambient container theorem that could combine with the paper's cubic-container result to prove the Big-Line–Big-Clique Conjecture. It is posed as an open problem.