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

Let $\ell$ and $k$ be positive integers satisfying

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

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

$$\operatorname{RR}(x+y+kz=\ell w)=\left\lfloor \frac{k+\ell+1}{\ell}\right\rfloor^2+O\left(\frac{k}{\ell^3}\right).$$

The paper proposes this modification in light of the results of Robertson and Myers and the exact values obtained in the paper. The original conjecture is partially false for $\ell=3$, so the remaining validity of this sharper error term is open.