A strengthening of the thrackle conjecture for topological trees

Let $\mathcal{P}$ be a set of $n$ points in the plane, and let there be $m$ topological trees such that every pair of trees intersects exactly once and every leaf of every tree belongs to $\mathcal{P}$. Strengthened thrackle conjecture. We conjecture that

$$m\le n.$$

This is posed as a strengthening of the thrackle conjecture, asserting a linear upper bound on the number of pairwise once-intersecting topological trees whose leaves lie in a prescribed set of points. The source does not indicate that this strengthening has been resolved.