Classification conjecture for mutually unbiased basis triplets in dimension six

Let $X_1,X_2,X_3$ be an MUB-triplet in dimension $d=6$, and let two MUB-triplets be permutationally unitary equivalent when they are related by the equivalence relation used in the construction referenced as $(\mathrm{hff})$. Let $X(\alpha)$ denote the Szöllősi family, let $F(x,y)$ denote the Fourier family, let $F^T(x,y)$ denote its transposed family, and let $F_6(1,1)$ denote the specified isolated Hadamard matrix. The transition matrices are

$$\sqrt{6}X_1X_2^*,\qquad \sqrt{6}X_2X_3^*,\qquad \sqrt{6}X_3X_1^*.$$

Classification conjecture. Any MUB-triplet $(X_1,X_2,X_3)$ in dimension $d=6$ is permutationally unitary equivalent to an MUB-triplet obtained by the construction $(\mathrm{hff})$. In particular, there are two essentially different cases: a $2$-parameter family of MUB-triplets, described in Szöllősi (year not given), in which one transition matrix belongs to the Szöllősi family $X(\alpha)$, one belongs to the Fourier family $F(x,y)$, and one belongs to the transposed family $F^T(x,y)$; or an isolated MUB-triplet in which all three transition matrices are equivalent to $F_6(1,1)$.

The conjecture is motivated by numerical experiments on $20000$ numerically obtained MUB-triplets: every one was, up to numerical tolerance, equivalent to a triplet from the stated construction. The numerical evidence does not establish the classification, so the general claim remains open.