The order of long rainbow arithmetic progressions

Let $T_k$ be the minimum positive integer $t$ such that, for every positive integer $n$, every equinumerous $t$-coloring of $[tn]$ contains a rainbow $k$-term arithmetic progression.

Jungić, Licht, Mahdian, Nešetřil and Radoičić conjectured that $T_k=\Theta(k^2)$, while Conlon, Fox and Sudakov proved that $T_k=O(k^2\log k)$.

We prove the matching lower bound $T_k=\Omega(k^2\log k)$, and hence $T_k=\Theta(k^2\log k)$.