The complex-subalgebra characterization conjecture for idempotent varieties

Let ${\mathcal V}$ be an idempotent variety satisfying the generalized entropic property. Let $\operatorname{\mathbf{CSub}}{\mathcal V}$ denote the corresponding complex-subalgebra variety.

Complex-subalgebra characterization conjecture.

$${\mathcal V}=\operatorname{\mathbf{CSub}}{\mathcal V}$$

if and only if ${\mathcal V}$ has a basis consisting of linear identities and the identities

$$f(x,\dots,x)=x$$

for every basic operation $f$.

The paper says that its characterization theorem does not decide this conjecture, which was attributed to earlier work; its status is therefore left open.