Linear-edge construction conjecture for sparse intersecting hypergraphs

For each positive integer $r$, let $f(r)$ be the minimum number of edges in an $r$-partite intersecting hypergraph $\mathcal{H}$ satisfying $\tau(\mathcal{H}) \geq r-1$.

Linear-edge construction conjecture. The function $f(r)$ grows linearly:

$$f(r)=\Theta(r).$$

The paper proves the lower bound $f(r)\geq (3-1/\sqrt{18})r(1-o(1))\approx 2.764r(1-o(1))$, but has no matching upper bound; the conjecture asks for constructions with a linear number of edges.