The growth-ratio conjecture for van der Waerden numbers in the number of colours

Let $W(r,k)$ denote the van der Waerden number, and write $W(r,k) = (r)^{n_r}$ and $W(r+1,k)=(r+1)^{n_{r+1}}$ for the corresponding exponents. Growth-ratio conjecture. As $r$ grows large with $r>k$,

$$\frac{W(r+1,k)}{W(r,k)}<r^{n_{r+1}-n_r}(r+1).$$

The conjecture proposes an upper bound for the ratio of successive van der Waerden numbers as the number of colours increases; the source gives no resolution.