Classification of the automorphic loops constructed from anisotropic planes

Let $p$ be a prime and let $F=GF(p)$. For $A,B\in GL(2,p)$, suppose that $FI\oplus FA$ and $FI\oplus FB$ are anisotropic planes. Let $Q(A)$ and $Q(B)$ be the loops constructed by the construction referred to in the source.

Classification conjecture. The loops $Q(A)$ and $Q(B)$ are isomorphic if and only if $A$ and $B$ are of the same type.

This asserts that the type of the defining matrices completely determines the isomorphism class among these constructed automorphic loops. The supplied source does not indicate whether the statement has been proved or disproved.