Suppose $A$ is a diagonal matrix with eigenvalues $1,\frac{1}{2},\frac{1}{3},\ldots,\frac{1}{n}$ and $x$ is drawn from standard Gaussian in $n$ dimensions. Define $z_n$ as follows $$z_n=E_{x\sim \mathcal{N}\left(0, I_n\right)}\left[\frac{x^T A^2 x}{x^T A^3 x}\right]$$

Is it possible to prove or disprove the following?

$$\lim_{n\to \infty} z_n = 2$$

This is a crosspost from math.SE where several people provided altnernative characterizations of $z_n$ but which don't quite settle the question.

Motivation: $z_n$ is the expected value of learning rate which maximizes loss decrease for a gradient descent step on a quadratic $A$ and random starting point. If the limit is 2, this would be a nice mathematical illustration behind the heuristic used in practice, "in high dimensions -- set learning rate as high as possible"

1more comment