The superlinear blocker conjecture with bounded collinearities

For fixed $\ell$, let $b_\ell(n)$ be the minimum integer such that every set of $n$ points in the plane with no $\ell$ collinear points can be blocked by a set of $b_\ell(n)$ points. Bounded-collinearity blocker conjecture. For all fixed $\ell$,

$$\frac{b_\ell(n)}{n}\rightarrow\infty\quad\text{as }n\rightarrow\infty.$$

This generalizes the general-position case $b_3(n)=b(n)$ and is stated in the source as implying the Big-Line-Big-Chromatic-Number Conjecture; its general status remains open.