How does the spectrum of a product of $k$ random matrices behave around 0?

In particular, I'm wondering if the CDF of squared singular values behaves as $x^{\frac{1}{k+1}}$ around 0. The result for $k=1$ comes from Marchenko-Pastur.

Below is a visualization of empirical CDF of squared singular values of a matrix formed by multiplying together $k$ square $1000\times 1000$ matrices with standard Normal entries.

[![enter image description here][1]][1]
[Notebook](https://www.wolframcloud.com/obj/yaroslavvb/nn-linear/forum-mo-products-of-matrices.nb)

Unclear from simulation above is whether apparent "faster than power-law" decay is a finite sample effect or if it persists for $n\to\infty$. Any tips?

Motivation: a potential candidate for RMT model to match scaling laws observed in neural network training ([background](https://random-matrix-learning.github.io/#presentation3)). Shape of loss curve over time for $t\to \infty$ comes from shape of spectrum for $x\to 0$


  [1]: https://i.sstatic.net/X469D.png