The parametrized Gompf-equivalence conjecture for Cappell–Shaneson matrices

Let

$$\mathcal{C}\mathcal{S}=\{(c,d,n)\in\mathbb{Z}^3\mid f_n(c)\equiv0\pmod d\ \text{and}\ d\neq0\}.$$

The set $\mathcal{C}\mathcal{S}$ parametrizes standard Cappell–Shaneson matrices via $(c,d,n)\mapsto X_{c,d,n}$. Let $\sim$ denote Gompf equivalence on $\mathcal{C}\mathcal{S}$, and let $(1,1,2)$ correspond to $A_0$. The parametrized Gompf-equivalence conjecture. For every $(c,d,n)\in\mathcal{C}\mathcal{S}$,

$$(c,d,n)\sim(1,1,2).$$

This is a reformulation of Gompf's matrix-equivalence conjecture in the parametrization by triples. The supplied text gives no separate resolution, so the general assertion remains open there.