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

Let $w(2;3,t)$ denote the van der Waerden number for 2-colourings with monochromatic progressions of lengths $3$ and $t$. Quadratic upper-bound conjecture. For $t\geqslant 3$,

$$w(2;3,t)\leqslant \frac{3}{2}t^2-\frac{3}{2}t.$$

This candidate occurs inside an ignored block in the supplied source context, whereas the active text gives the weaker conjecture with an unspecified constant $c>1$; it is therefore retained separately only because the parser supplied it as a stated conjecture span.