Minority-pair conjecture for minimal bounded width algebras

Let $\mathbb{A}$ be a minimal bounded width algebra, and let $a,b\in\mathbb{A}$. Write $\operatorname{Sg}_{\mathbb{A}^2}\{(a,a),(a,b),(b,a)\}$ for the subalgebra of $\mathbb{A}^2$ generated by these three pairs. Write $a\mathbin{\rightarrow}b$ for the directed relation used in the paper.

Minority-pair conjecture. If

$$(b,b)\in\operatorname{Sg}_{\mathbb{A}^2}\{(a,a),(a,b),(b,a)\},$$

then $a\mathbin{\rightarrow}b$.

This would imply that minimal bounded width algebras have no minority pairs. The conjecture holds in every majority algebra and every spiral, but its general status is unresolved in the supplied text.