The quasi-arithmetic obstruction to Poissonian pair correlations

Let $(a(n))_{n\geq1}$ be a strictly increasing sequence of positive integers. Call it quasi-arithmetic if it has the finite-dimensional arithmetic-progression structure defined in the source. For a real number $\alpha$, consider $(\{a(n)\alpha\})_{n\geq1}$. Quasi-arithmetic obstruction conjecture. If $(a(n))_{n\geq1}$ is quasi-arithmetic, then there is no $\alpha$ such that the pair correlations of $(\{a(n)\alpha\})_{n\geq1}$ are asymptotically Poissonian. This is presented as an equivalent formulation of the preceding additive-energy conjecture.