On the equivalence of certain quasi-Hermitian varieties

In [A. Aguglia, A. Cossidente, G. Korchmaros, "On quasi-Hermitian varieties", J. Comb. Des. 20 (2012), 433-447] new quasi-Hermitian varieties ${\mathcal M}_{\alpha,\beta}$ in $\mathrm{PG}(r,q^2)$ depending on a pair of parameters $\alpha,\beta$ from the underlying field $\mathrm{GF}(q^2)$ have been constructed. In the present paper we determine the projective equivalence classes of such varieties for $r=3$ and $q$ odd.