Pinnable rainbow matching conjecture

Let $\mathcal{H}=(H_1,\ldots,H_m)$ be a family of $r$-uniform hypergraphs. A rainbow matching is a matching containing at most one edge from each $H_i$, and $\nu_R(\mathcal{H})$ is its maximum size. Assume that $\bigcup_{i=1}^mH_i$ is pinnable, meaning that some set meets every edge of the union in exactly one vertex.

Pinnable rainbow matching conjecture. If

$$\tau\left(\bigcup_{i\in I}H_i\right)>(2r-2)(|I|-1)$$

for every $I\subseteq[m]$, then $\mathcal{H}$ has a full rainbow matching:

$$\nu_R(\mathcal{H})=m.$$

This strengthens the earlier rainbow matching conjecture by replacing the common partite structure with pinnability. It remains open.