Conjecture on isomorphism types of commutative A-loops of order $p^3$

Let $p>3$ be a prime. Let $a_1 e 0$ be a quadratic residue modulo $p$ and let $a_2 e 0$ be a quadratic nonresidue modulo $p$. For the loop construction denoted by $\operatorname{terg}{\mathbb Z_p}{a}{b}$, consider the loops with parameters $(a_1,0)$ and $(a_2,0)$.

Isomorphism-type conjecture. The loops

$$\operatorname{terg}{\mathbb Z_p}{a_1}{0}$$

and

$$\operatorname{terg}{\mathbb Z_p}{a_2}{0}$$

are not isomorphic.

The conjecture concerns whether the quadratic-residue and quadratic-nonresidue parameter classes remain distinct in the classification for primes $p>3$. The authors state that they did not manage to establish it, and the source gives no resolution.