On Salem numbers which are exceptional units II

We show that for any natural number $n$ satisfying $n\equiv 4\mathrm{m}\mathrm{o}\mathrm{d}8$ and $n\equiv ̸0\mathrm{m}\mathrm{o}\mathrm{d}5$, and for any odd integer $t\ge \frac{n+6}{2}$ there are infinitely many Salem numbers $\alpha$ of degree $2t$ such that ${\alpha }^n-1$ is a unit. This result, obtained using a generalization of a construction due to Gross and McMullen [5], partially completes the main result of [7].