Four-variable Rado number conjecture for $x+a(x+y)=2aw$

Let $a$ be a positive integer, and let $R(x+a(x+y)=2aw)$ denote the 2-color Rado number of the four-variable equation $x+a(x+y)=2aw$. Four-variable Rado number conjecture.

$$R(x+a(x+y)=2aw)=a^2.$$

The conjecture proposes an extension of the previously determined three-variable equation $x+ay=2az$; the supplied text gives no resolution evidence.