The two-element projection quotient conjecture for clone-minimal dispersive algebras

Let $\mathbb{A}$ be a clone-minimal dispersive algebra. For elements $a,b e a$ of $\mathbb{A}$, write $\operatorname{Sg}\{a,b\}$ for the subalgebra they generate, $D_{\operatorname{Sg}\{a,b\}}$ for its associated digraph, and $O(a)$ and $O(b)$ for the corresponding sets. A two-element projection algebra is the algebra on two elements whose basic operations are projections.

Two-element projection quotient conjecture. For every $a\ne b\in\mathbb{A}$, there is a surjective homomorphism

$$\operatorname{Sg}\{a,b\}\twoheadrightarrow P,$$

where $P$ is a two-element projection algebra. Equivalently, $D_{\operatorname{Sg}\{a,b\}}$ has exactly two weakly connected components, or

$$O(a)\cap O(b)=\emptyset$$

in $\operatorname{Sg}_{\mathbb{A}}\{a,b\}$.

The paper presents this as a difficult conjecture about clone-minimal dispersive algebras; no resolution is supplied in the source.