A quadratic upper-growth conjecture for the van der Waerden numbers $w(2;3,t)$

Let $w(2;3,t)$ denote the least integer $N$ such that every 2-colouring of $\{1,\ldots,N\}$ contains a monochromatic arithmetic progression of length $3$ in one colour or length $t$ in the other. Quadratic upper-growth conjecture. There exists a constant $c>1$ such that

$$w(2;3,t)\leqslant ct^2.$$

The precise bound $w(2;r,t)\leqslant t^2$ has been invalidated in the paper, but the authors state that quadratic growth still seems appropriate; the conjecture remains open in the supplied text.