Gompf's conjecture on Cappell–Shaneson matrix equivalence

A Cappell–Shaneson matrix is a matrix $A\in SL(3;\mathbb{Z})$ satisfying $\det(A-I)=1$. Two such matrices are Gompf equivalent when they are related by Gompf's equivalence relation on Cappell–Shaneson matrices; let $A_0$ denote the matrix $X_{1,1,2}$. Gompf's matrix-equivalence conjecture. Every Cappell–Shaneson matrix is Gompf equivalent to $A_0$. If true, this would reduce the study of the corresponding Cappell–Shaneson spheres to the standardness of the sphere associated with $A_0$. Gompf proved the conjecture for the traces $-6\leq n\leq 9$ and for $n=11$, while the general statement remains open in the supplied text.