The MEX property for groups in $On_p$

Let $On_p$ be the class of ordinals equipped with the paper's addition and multiplication, and let $\alpha,\beta\in On_p$ be groups. The unordered pair $\{\alpha,\beta\}$ has the MEX property when

$$\alpha\beta=\operatorname{mex}\{\alpha'\beta+\alpha\beta'-\alpha'\beta': \alpha'<\alpha,\beta'<\beta\}.$$

MEX conjecture. If $\alpha,\beta\in On_p$ are groups, then the pair $\{\alpha,\beta\}$ has the MEX property.

The MEX property would provide a multiplicative characterization for pairs of group ordinals, despite the fact that the analogous definition fails for general elements of $On_p$ when $p\geq 3$. The source gives no resolution of this conjecture.