The minor-homomorphism conjecture for totally symmetric and Mal'cev clones

Let $E_k$ be a $k$-element set with $k\geq 2$, and let $\mathcal S$ be a clone over $E_k$. Say that $\mathcal S$ satisfies $\operatorname{TS}(n)$ for every $n\geq 2$ and satisfies the Mal'cev condition $\Sigma_{\operatorname{M}}$. The minor-homomorphism conjecture. There exists a minor homomorphism from the clone $\mathcal I_2$ to $\mathcal S$. This conjecture identifies the presence of all totally symmetric identities together with the Mal'cev condition as sufficient for a minor homomorphism from $\mathcal I_2$; the excerpt gives no resolution.