Parity conjecture for the Rado number of $x+qy=q^2z$

Let $q\in\mathbb{N}$, and let $R(x+qy=q^2z)$ denote the 2-color Rado number of the equation $x+qy=q^2z$. Parity conjecture.

$$R(x+qy=q^2z)= \begin{cases} \frac{q^3}{2} & \text{if } q\equiv 0\bmod 2,\\ \frac{q^3+q}{2} & \text{if } q\equiv 1\bmod 2. \end{cases}$$

The claim is presented as an empirical conjecture based on Rado numbers computed using the authors' algorithm; the supplied text gives no evidence that it has been resolved.