The quadratic growth-ratio conjecture for van der Waerden numbers in the progression length

Let $W(r,k)$ denote the van der Waerden number. Quadratic growth-ratio conjecture. As $k$ grows large with $k>r$,

$$\frac{W(r,k+1)}{W(r,k)}<k(k+1).$$

This is the simplified form obtained in the source under the additional assumption that $m_{k+1}-m_k\in[0,1]$; the source gives no resolution.