The irrational-angle counterexample conjecture for weak rotational symmetry

Let $K$ be the compact group acting by conjugation, let $\theta_0$ be the corresponding group element, and let $A$ be a matrix satisfying the discrete rotational-symmetry relation

$$\mathrm{e}^{\mathrm{i}k\varphi_0}A=\mathrm{e}^{\Omega_0 k}A\mathrm{e}^{-\Omega_0 k}$$

for all $k\in\mathbb{Z}$ and some $\Omega_0\in\mathfrak{k}$. A counterexample means that this relation holds while the continuous relation fails for some real $t$.

Irrational-angle counterexample conjecture. It is impossible to find a counterexample satisfying $\varphi_0\notin 2\pi\mathbb{Q}$.

The conjecture concerns the passage from discrete to continuous rotational symmetry and would require deeper results on the structure of the torus algebra and its root-space decomposition of $K$.