The Wigner semicircle law states that for a random GOE-matrix $M^N \in \mathbb{R}^{N \times N}$ in the $N \rightarrow \infty$ limit for any $f \in C^b(\mathbb{R})$

$$\lim_{N \rightarrow \infty}\frac{1}{N} \mathbb{E}^{\text{GOE}}\left(\sum_{\lambda \text{ eigenvalue of } M^N} f(\lambda)\right) = \int f(\lambda) \rho(\lambda) d\lambda$$

where $\rho$ is the semicircle distribution.

Now, I am in the situation that I would like to compute

$$\lim_{N \rightarrow \infty}\frac{1}{N} \mathbb{E}^{\text{GOE}}\left(\sum_{\lambda \text{ eigenvalue of } M^N} 1_{\lambda_0>x}f(\lambda)\right) $$

where $\lambda_0$ is the smallest eigenvalue of $M^N$.

My question is: Is there still a limiting distribution?

EDIT: It is mentioned in the comments by Carlo Beenakker that the lowest eigenvalue accumulates with high probability at $-R$ where $[-R,R]$ is the support of the semicircle distribution and thus it should follow that $$\lim_{N \rightarrow \infty}\frac{1}{N} \mathbb{E}^{\text{GOE}}\left(\sum_{\lambda \text{ eigenvalue of } M^N} 1_{\lambda_0>x}f(\lambda)\right) = 1_{\{x\le -R\}} \int f(\lambda) \rho(\lambda) d\lambda.$$

Thus, the question is whether one can show that this limit really exists (i.e. that one can combine the accumulation of the lowest eigenvalue and the convergence to the semicircle distribution as shown above).