Binary-invariants conjecture for minimal bounded width algebras

Let $\mathbb{A}$ be a minimal bounded width algebra. Define

$$\operatorname{Inv}_2(\mathbb{A})$$

to be the collection of subalgebras of $\mathbb{A}\times\mathbb{A}$. Two algebras are term equivalent when they have the same term operations, up to the corresponding choice of basic operations.

Binary-invariants conjecture. Among minimal bounded width algebras, $\mathbb{A}$ is determined up to term equivalence by $\operatorname{Inv}_2(\mathbb{A})$.

The conjecture would say that the binary subalgebra structure completely determines a minimal bounded width algebra up to term equivalence. The supplied text gives no evidence of a proof or counterexample.