Robertson–Myers conjecture for the Rado number of x+y+kz=ℓw

Let $cell$ and $k$ be positive integers satisfying

$$\ell\geq 2,\qquad k\geq \ell+2.$$

For the 2-color Rado number $\operatorname{RR}(x+y+kz=\ell w)$, Robertson–Myers conjecture. for fixed $\ell$,

$$\operatorname{RR}(x+y+kz=\ell w)=\left\lfloor \frac{k+\ell+1}{\ell}\right\rfloor^2+O\left(\frac{k}{\ell^2}\right).$$

This conjecture was proposed by Robertson and Myers for the case $m=4$. Saracino and Wynne showed that it is partially false when $\ell=3$, while determining the exact 2-color Rado number in that case.