The conjecture that half-isomorphisms of automorphic loops are special

Let $Q$ and $P$ be automorphic loops, and let $f:Q\to P$ be a half-isomorphism, meaning that for all $x,y\in Q$,

$$f(x*y)=f(x)*f(y)\quad\text{or}\quad f(x*y)=f(y)*f(x).$$

Special half-isomorphism conjecture. Every half-isomorphism between automorphic loops is special.

The paper proves this assertion for automorphic loops satisfying an additional identity, including automorphic Moufang loops, and notes that known nontrivial examples of half-isomorphisms between automorphic loops are all special. The conjecture proposes that this holds for all automorphic loops.