Sharpness conjecture for Orthomin(k) on root-of-unity diagonal systems

Let $k\in\mathbb{N}$. For each $d\in\mathbb{N}$, let $U=\operatorname{diag}(1,\zeta_d,\zeta_d^2,\dots,\zeta_d^{d-1})$, where $\zeta_d=\exp(2\pi \mathbf{i}/d)$ is a primitive root of unity of order $d$. For $0<\rho<1$, consider the linear system from the source with matrix $U$ and initial value $x_0=0$, and let $r_n^{(k)}$ denote the residuals produced by Orthomin$(k)$. Sharpness conjecture. For every $k\in\mathbb{N}$, there exist $d_k\in\mathbb{N}$ and $0<\rho_k<1$ such that, for every $\rho\in(0,\rho_k)$ and every $d\ge d_k$,

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

The result is proved in the paper for $k=1$, while for $k\geq 2$ the authors report strong numerical evidence. If true, it shows that Orthomin$(k)$ need not converge asymptotically faster than Orthomin$(1)$ in general.