Rado-number formula for ${\cal E}(3,0;a,a,b)$

Let $a$ and $b$ be coprime positive integers, with $a$ odd, and suppose that

$$3\leq a<b\leq 2a-1.$$

Rado-number formula. Then

$$R_3({\cal E}(3,0;a,a,b))=a^3b.$$

The conjecture is based on computational data for Rado numbers of the equation ${\cal E}(3,0;a,a,b)$; no proof or resolution is supplied in the source.