Minimal Taylor algebra structural conjecture

Let $\mathbb{A}$ be a minimal Taylor algebra, meaning a Taylor algebra whose clone is minimal in the sense used by the source, and suppose that $\mathbb{A}$ is generated by two elements $a,b\in\mathbb{A}$. A ternary absorbing subalgebra $\mathbb{C}\lhd_Zmathbb{A}$ is a proper subalgebra with the ternary absorption property denoted by $lhd_Z$ in the source. Minimal Taylor structural conjecture. At least one of the following holds: $\mathbb{A}$ has a congruence quotient that is an affine algebra of prime order; it has a congruence quotient that is a two-element semilattice; it has a congruence quotient that is a two-element majority algebra; or there are proper ternary absorbing subalgebras $\mathbb{C},\mathbb{D}\lhd_Zmathbb{A}$ such that $a\in\mathbb{C}$, $b\in\mathbb{D}$, $\mathbb{C}\cupmathbb{D}=\mathbb{A}$, and $\mathbb{C}\capmathbb{D}\neq\varnothing$. The conjecture is presented as unresolved and would provide a unified structural description.