RPCD worst-case conjecture for positive-definite quadratics

Let $n$ be the dimension, let ${\bm A}\in\mathbb{S}_{+}^{n}$ be a positive-definite quadratic Hessian with $\lambda_{\min}({\bm A})=\sigma\in(0,1]$, and let ${\bm x}_0\in\mathbb{R}^{n}\setminus\{0\}$. If ${\bm x}_K$ denotes the output of random-permutation coordinate descent (RPCD) after $K$ epochs, then

$$\lim_{K\to\infty}\left(\frac{\mathbb{E}\left[\|{\bm x}_K\|^2\right]}{\|{\bm x}_0\|^2}\right)^{1/K}\leq\max\left\{\left(1-\frac{1}{n}\right)^n,\left(1-\frac{\sigma}{n}\right)^{2n}\right\}.$$

RPCD worst-case conjecture. The upper bound above, proved for the specified permutation-invariant class of quadratic Hessians, extends to all positive-definite quadratic functions. This would imply that the class contains worst-case examples for RPCD among all positive-definite quadratics. The conjecture is motivated by numerical searches and is presented as an open extension beyond the class analyzed in the paper.