The bounded-intersection hub curve conjecture

Let $\mathcal{F}$ be a family of curves in the plane, and let $t$ be a positive integer. Suppose that one curve in $\mathcal{F}$ intersects every other curve in $\mathcal{F}$ in at least one and at most $t$ points.

Bounded-intersection hub curve conjecture. For every $t$, the intersection graph of $\mathcal{F}$ is $\chi$-bounded.

The corresponding statement with exactly one intersection point is known, and the conjectured bounded-intersection extension would imply an O(n\begin{math}\log n\end{math}) edge bound for certain quasi-planar topological graphs. The source does not provide a resolution.