Recursive subalgebra conjecture for minimal bounded width algebras

Let $\mathbb{A}$ be a minimal bounded width algebra and let $a,b\in\mathbb{A}$. Denote by $\operatorname{Sg}\{a,b\}$ the subalgebra generated by $a$ and $b$.

Recursive subalgebra conjecture. The set

$$\operatorname{Sg}\{a,b\}\setminus\{a\}$$

is a subalgebra of $\mathbb{A}$.

This is proposed as an analogue of the paper's recursive theorem for general bounded width algebras. If true, it would support an inductive classification of minimal bounded width algebras generated by two elements; the supplied text gives no resolution.