Small-gap limiting law for Gaussian beta ensembles

Let $\beta>0$, and let the Gaussian beta ensemble G$\beta$E have joint density

$$\frac{1}{Z_{n,\beta}}e^{-n\beta \sum_{i=1}^n\lambda_i^2/2}\prod_{1\leq i<j\leq n}|\lambda_i-\lambda_j|^\beta.$$

Let $t^n_{k,\beta}$ be its $k$-th smallest gap. Small-gap conjecture. There exists a constant $c_\beta$ depending on $\beta$ such that

$$\tau^n_{k,\beta}=c_\beta n^{(\beta+2)/(\beta+1)}t^n_{k,\beta}$$

has limiting density

$$\frac{\beta+1}{(k-1)!}x^{k(\beta+1)-1}e^{-x^{\beta+1}}$$

as $n\to\infty$. This conjecture extends the observed smallest-gap laws for the Gaussian orthogonal and unitary ensembles, and proposes the same limiting behavior as for circular beta ensembles.