Linear growth conjecture for rainbow matching thresholds

Let $f(r,t)$ be the largest number of colors for which there exists an $(f,t)$-colored $r$-partite $r$-graph without a rainbow $t$-matching. Linear growth conjecture. For every $r$ there exists a constant $c_r$ such that

$$f(r,t)\leq c_r t$$

for all $t$. This conjecture proposes the correct order of growth in $t$ after the Aharoni–Berger conjecture was refuted. The paper proves an exponential upper bound, but the claimed linear bound remains open.