An improved lower bound for the van der Waerden number $w(3,k)$

For an integer $k\ge 3$, let $w(3,k)$ be the least $n$ such that every red-blue coloring of $[n]$ contains either a nontrivial three-term arithmetic progression in blue or a nontrivial $k$-term arithmetic progression in red. A recent breakthrough of Green proved that $w(3,k)\ge k^{Ω\left(\left(\frac