Shelah's automorphism conjecture for free algebras

Let $F$ be freely generated by $\\{x_i:i<\omega+\omega\\}$, and let $\langle x_i:\omega\leq i<\omega+\omega\rangle_F$ denote the subalgebra generated by those elements. Shelah's automorphism conjecture. For varieties satisfying

$$(*)\quad \text{there is an automorphism }h\text{ of }F\text{ with }h(x_i)=x_i\text{ for }i<\omega\text{ and }h(x_\omega)\notin\langle x_i:\omega\leq i<\omega+\omega\rangle_F,$$

a similar answer holds for the automorphism group, whereas if $(*)$ fails the group is like the permutation group of $\lambda$.

This is stated as a belief about the automorphism group of a free algebra; the supplied text gives no resolution.