Poisson point-process and Gumbel conjecture for the low-lying eigenvalues

Let $d\in \\{1,2,3\\}$. Define

$$a_L:= \big(C_d \log L\big)^{\frac1{2-\frac{d}{2}}},\quad b_L:= \frac{C_d}{d (2-\frac{d}{2}) a_L^{1-\frac{d}{2}}}.$$

Poisson point-process and Gumbel conjecture. The point process

$$\left(\frac{\lambda_{n,L} + a_L}{b_L}\right)_{n\ge 1}$$

converges in law as $L\to\infty$ to a Poisson point process on $\mathbb R$ of intensity $e^x dx$. In particular,

$$-\frac{\lambda_{1,L}+a_L}{b_L}$$

converges in law to a Gumbel random variable.

The preceding tail estimates show concentration of the low-lying eigenvalues near the critical scale $a_L$; this conjecture predicts the finer limiting point-process fluctuations and the associated Gumbel law for the lowest eigenvalue. The source does not state a resolution of this conjecture.