Almost-everywhere obstruction conjecture for Poissonian pair correlations

Let $(a_n)_{n\in\mathbb N}$ be a sequence of distinct integers, and consider the sequence $((\lbrace a_n\alpha\rbrace)_{n\in\mathbb N})$ for real $\alpha$. Say that this sequence has Poissonian pair correlations when its pair-correlation statistics are Poissonian.

Almost-everywhere obstruction conjecture. If for almost all $\alpha$ the pair correlations of $((\lbrace a_n\alpha\rbrace)_{n\in\mathbb N})$ are not Poissonian, then the pair correlations of this sequence are not Poissonian for any $\alpha$.

This conjecture asserts that failure of Poissonian pair correlations for almost every parameter cannot coexist with Poissonian pair correlations for even one parameter. The surrounding results establish strong obstructions when the additive energy of the initial segments is maximal, but the stated implication itself is presented as a conjecture.