Infinite-dimensional sharpness conjecture for Orthomin(k)

Let $\mathbb{S}^1$ be the unit circle with Haar probability measure $d\mu_0$, and let $U:L^2(\mathbb{S}^1,d\mu_0)\to L^2(\mathbb{S}^1,d\mu_0)$ be the multiplication operator

$$Uf(z)=zf(z).$$

Consider the linear system from the source with right-hand side $b(z)=1$ for all $z\in\mathbb{S}^1$ and zero initial guess, and let $r_n^{(k)}$ be the residuals produced by Orthomin$(k)$. Infinite-dimensional sharpness conjecture. For every $0<\rho<1$ and every $k\in\mathbb{N}$,

$$\lim_{n\to\infty}\frac{\lVert r_{n+1}^{(k)}\rVert}{\lVert r_n^{(k)}\rVert}=\rho.$$

This is the infinite-dimensional analogue of the finite root-of-unity construction and is presented as a conjectural extension for all $k$; the supplied text gives no proof or resolution.