Poissonian correlation conjecture for fractional parts of $\alpha n^\theta$

Let $m\ge 2$, let $\alpha>0$, and let $0<\theta<1/m$. Consider the sequence of fractional parts $(\alpha n^\theta\bmod 1)_{n>0}$. Poissonian correlation conjecture. For every such $m$, $\alpha$, and $\theta$, this sequence has Poissonian $m$-point correlation. The conjecture would extend the paper's theorem to the natural boundary $\theta<1/m$; the stated theorem only reaches a smaller range because of the available exponential-sum and oscillatory-integral estimates, and proving the full range would require significant new ideas.