pr.probability – Average value of $frac{x’A^2x}{x’A^3x}$ over surface of $n$-dimensional sphere

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
$$z_n=E_{xsim mathcal{N}left(0, I_nright)}left(frac{x^T A^2 x}{x^T A^3 x}right)$$

Is it possible to prove or disprove the following?

$$lim_{nto infty} z_n = 2$$

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”