Poissonian gap distribution conjecture for $\alpha n^\theta$

Let $\alpha\neq 0$ and let $\theta>0$ satisfy $\theta\neq\tfrac12$ and $\theta\notin\mathbb Z$. For each $N$, let $y_1^{(N)}<\cdots<y_N^{(N)}$ be an ordering of the points $\alpha n^\theta\pmod 1$ with $n<N$. Poissonian gap distribution conjecture. The normalized consecutive gaps satisfy

$$\lim_{N\to\infty}\frac{1}{N}\#\left\{i<N:N\bigl(y_{i+1}^{(N)}-y_i^{(N)}\bigr)\in(a,b)\right\}=\int_a^b e^{-t}\,dt.$$

In other words, the gap distribution should be Poissonian. The case $\theta=\tfrac12$ is excluded because the gap distribution is known not to be Poissonian there; this is also the only case in which existence of the gap distribution is known.